i 


EGINNERS 
GEBRA 

SYKES 
COMSTOCK 


GIFT  or 

Publisher 


EDUCATION  DEPT. 


BEGINNERS'  ALGEBRA 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/beginnersalgebraOOcomsrich 


BEGINNERS'  ALGEBRA 


By 
MABEL  SYKES 

Instructor  in  Mathematics,  Bowen  High  School,  Chicago 

Author  of  ''A  Source  Book  of  Problems  for  Geometry'^ 

and  {with  Clarence  E.  Comstock)   of  ''Plane 

and  Solid  Geometry'' 


and 


CLARENCE  E.  COMSTOCK 

Professor  of  Mathematics,  Bradley  Polytechnic  Institute, 
Peoria,  Illinois 


RAND  M9NALLY  &  COMPANY 

CHICAGO  NEW  YORK 


1^* 


Copyright,  1922,  by 
Rand  McNally  &  Company 


^ 


Mad.- in  us.  A. 


B-23 


THE   CONTENTS 

PAGB 

The  Preface vii 

CHAPTER 

I.  Introductory:  Fractions  and  Decimals  of  Arith- 
metic      . 1 

Fractions;  decimals;  approximate  numbers;  order  of 
operations 

II.  Formulas  and  Equations     . 20 

Formulas;  operations  on  equations  leading  to  their 
solution 

III.  The  Solution  of  Problems 42 

Problems  giving  numerical  equations;  problems  giving 
general  equations;   evaluation  of  formulas 

IV.  Negative  Numbers 65 

Meaning;  addition;  subtraction;  multiplication;  divi- 
sion; equations  and  problems 

V.  Graphics 101 

Bar  diagrams ;  line  diagrams ;  graphic  solution  of  prob- 
lems; graphs  of  algebraic  expressions  and  equations 

VI.  Linear  Equations  in  Two  Unknowns 123 

Graphic  solution;  algebraic  solution;  problems 

VII.  Special  Products;  Factoring;  Equations  Solved 

BY  Factoring 137 

Products  and  factors;  common  factor  type;  graphs; 
trinomial  type;  equations  and  problems;  square  of  bino- 
mial ;  product  of  sum  and  difference ;  summary ;  equations 
and  problems 

VIII.  Review  and  Extension  of  Fundamental  Opera- 
tions        181 

Definitions;  addition;  multiplication ;  subtraction ;  divi- 
sion; linear  equations  in  one  unknown;  factorable  equa- 
tions in  one  unknown;  sets  of  linear  equations  in  two 
unknowns;    the  stating  of  problems 

V 


vi  THE  CONTENTS 

PAGE 

IX.  Fracxcions 208 

Reduction;  multiplication;  division;  addition;  complex 
fractions;  graphs;  ratio;  variation 

X.  Square  Roots  and  Quadratic  Equations     .     .     .  229 

Square  root ;  solution  of  quadratic  equations  by  square- 
root  method;  graphs;  problems 

XL  Fractional  Equations  in  One  Unknown     .     .     .  245 
Solutions;  transformation  of  formulas;  problems 

XII.  Sets  of  Equations  in  Two  Unknowns    ....  257 
Introduction;  two  equations  of  the  first  degree;  frac- 
tional equations;  one  equation  of  first  and  one  of  second 
degree;  graphs;  problems 

XIII.  Radicals 272 

Definitions;  addition;    multiplication;   division;   irra- 
tional equations 

The  Appendix 

A.  Factoring 287 

Sum   of  two  cubes;    difference  between   two  cubes; 
factoring  by  grouping  after  expanding 

B.  Square  Roots 291 

Square  root  of  polynomials;  square  root  of  arithmeti- 
*  cal  numbers 

C.  Exponents 296 

Negative;  fractional;  zero 

D.  Table  of  Square  Roots 298 

The  Index 299 


THE   PREFACE 

This  book  is  the  first  of  a  two-book  series.  It  has  been 
the  intention  of  the  authors  to  include  in  Beginners*  Algebra 
no  more  work  than  can  be  completed  easily  in  one  year. 
It  is  believed  that  after  this  course  has  been  finished  the 
pupil  will  have  a  clear  understanding  of  the  principles  pre- 
sented and  the  ability  to  apply  them  to  algebraic  expression 
of  less  complicated  forms. 

The  order  in  which  some  of  the  topics  occur  may  be  a 
little  unusual ;  yet  it  is  felt  that  the  pupil  will  find  his  progress 
is  easy  and  invigorating.  The  subject  is  presented  in  a 
manner  to  stimulate  the  power  of  the  pupil  to  think  in  mathe- 
matical terms.  It  is  hoped  that  the  language  is  so  simple 
and  direct  that  the  pupil  will  find  little  or  no  difficulty  in 
reading  the  book  with  understanding  and  pleasure.  Tech- 
nical language  has  been  reduced  to  a  minimum. 

In  view  of  the  fact  that  many  high-school  pupils  have 
diflBictilty  in  using  fractions  and  decimals  with  speed  and 
accuracy,  an  introductory  chapter  is  devoted  to  such  matters. 
In  this  chapter  attention  is  also  called  to  the  proper  nvunber 
of  figures  to  be  retained  when  calculating  with  approximate 
numbers. 

The  close  connection  between  algebra  and  arithmetic  is 
stressed  throughout  the  book.  The  idea  that  letters  stand 
for  numbers  is  brought  to  the  surface  of  the  pupil's  mind  by 
frequent  exercises  in  the  evaluation  of  algebraic  expressions. 

The  equation  as  the  means  of  solving  problems  has  been 
taken  for  central  consideration.  The  methods  of  solving 
equations  are  developed  very  gradually  in  connection  with 
problems  giving  rise  to  them.  These,  as  well  as  other  prin- 
ciples and  methods,  receive  a  gradual  development  at  first 
from  a  more  or  less  common-sense  point  of  view,  followed 
by  a  more  formal  presentation. 

vii 


vni  THE  PREFACE 

Several  features  of  the  book  should  receive  special  men- 
tion. 

A  large  number  of  simple  exercises  are  given.  More  com- 
plicated and  difficult  expressions  are  left  to  the  later  coiu*se. 
This  is  notably  true  in  the  case  of  fractions,  factoring,  and 
radicals.  Certain  types  often  given  in  a  first-year  course 
are  omitted,  including  the  sum  and  difference  of  cubes  and 
forms  factored  after  grouping  terms.  These  have  been 
placed  in  the  Appendix  for  the  convenience  of  teachers  who 
wish  to  include  them  in  the  one-year  course. 

It  is  in  the  stating  of  problems  that  most  pupils  find  their 
greatest  difficulty.  The  problems  in  this  book  have  been 
chosen  from  topics  with  which  the  average  pupil  has  con- 
siderable familiarity.  The  approach  to  problems  of  the 
various  types  is  made  easy.  General  rules  of  procedure 
are  given  which  it  is  hoped  will  be  found  to  be  very  helpful. 

The  graph  is  considered  of  so  much  importance  that  it  is 
made  an  essential  part  of  the  course.  Many  ways  in  which 
graphs,  both  statistical  and  algebraic,  may  be  used  are 
indicated.  Place  is  given  to  the  solution  of  problems  by 
graphic  methods. 

Although  the  word  "function"  does  not  appear  in  the 
book,  the  idea  is  frequently  present,  notably  in  connection 
with  the  graph  which  is  called  for  when  various  algebraic 
forms  are  considered. 

The  important  distinction  between  equations  of  the  first 
and  second  degrees  and  fractional  equations  is  taken  into 
account,  and  a  separate  treatment  is  given  to  the  solution 
of  fractional  equations. 

It  is  believed  that  the  book  is  in  close  accord  with  the 
country-wide  movement  for  the  improvement  of  the  teach- 
ing of  algebra  to  those  beginning  the  study  of  the  subject. 

C.  E.  C. 
January,  ig22  M.  S. 


BEGINNERS'    ALGEBRA 


CHAPTER   I 

Introductory:  Fractions  and  Decimals  of 
Arithmetic 

1.  Equal  fractions.  Certain  fractions  that  look  very 
unlike  have  the  same  value,  ff ,  -y-,  and  f  are  equal  in 
value  though  they  differ  in  form.  We  may  reduce  ^  to 
the  form  f  by  dividing  both  terms  by  the  number  7. 

To  reduce  a  fraction  to  lower  terms  means  to  divide  both 
terms  of  the  fraction  by  2  or  3  or  some  larger  nimiber. 

A  fraction  is  said  to  be  in  its  lowest  terms  when  the  numer- 
ator and  denominator  cannot  be  divided  by  2  or  3  or  some 
larger  whole  number  without  a  remainder. 

We  may  reverse  this  operation  and  reduce  a  fraction  to 
higher  terms  if  we  so  desire.  We  may  multiply  both  terms 
of  f  by  13  and  get  ff •  To  reduce  f  to  a  fraction  whose 
denominator  is  12,  we  multiply  both  terms  by  4,  a  number 
which  multipUed  by  3  gives  12;  the  resulting  fraction  is  ■^. 

For  some  purposes  it  is  desirable  to  reduce  fractions  to 
their  lowest  terms,  and  for  other  purposes  it  will  be  fotmd 
necessary  to  reduce  them  to  higher  terms.  One  very 
important  fact  must  be  thoroughly  fixed  in  mind:  If  both 
terms  of  a  fraction  are  divided  (or  multiplied)  by  the  same 
number,  the  value  of  the  fraction  is  unchanged.  This  may 
be  called  the  Fraction  Law. 

EXERCISES 

Reduce  to  lower  terms: 

126154524  o^   —  ?5^?1 

18'  9'  30'  81'  48  *  25'  63'  9'  51'  30 


2         ■    '    ■'  '    '     BEGINNERS'  ALGEBRA 

■;  '    ,27;  5^,  ;30  ,m    126 
51'  54'  75'  33 '  168 

4.  Reduce  to  fractions  having  denominators  indicated: 

r.     ^-  3     7    13    3     5 

Fraction  -,    -,  -,  -,    - 

Denominator  21,  40,  45,  32,  63 

T,     ^-  2     3     3     2     1 

Fraction  -,    -,    -,    -,    - 

Denominator  12,  12,  70,  70,  70 

5.  Find  all  the  fractions  with  denominators  less  than  100  which 
are  equal  to  -f^. 

2.  To  multiply  together  an  integer  and  a  fraction. 

2^7^2X7^14  . 

3^  3         3 

Rule.    Multiply  the  numerator  of  the  fraction  by  the 
integer. 

Notice  the  very  important  special  case  in  which  the 
integer  and  the  denominator  are  the  same: 

|X3  =  2 

In  this  case  the  product  is  the  numerator.      The  rule 
would  call  for  work  like  this: 

But  the  work  is  done  more  quickly  thus: 

3X3--^  -2 

or,  better  still* 

|X^=2 

It  is  foolish  to  do  unnecessary  multiplications  such  as: 

ix«=f-"* 


INTRODUCTORY  3 

It  is  much  better  to  work  as  follows : 

3 

Whether  the  result  should  be  left  in  the  form  of  -^  or 
reduced  to  the  form  Yl\  or  to  the  form  17.33+  depends 
upon  what  use  is  to  be  made  of  it.  As  an  answer  17J  lb. 
is  preferable  to  ^  lb.  $17.33  is  better  than  -^^  dollars, 
but  in  computation  it  is  often  more  convenient  to  use  the 
form  ^. 

EXERCISES 

Find  products: 

1.7x|9x^,3x|  2.  7xf,  8x^9x1 

3.  15  x|  25x1  ^X49  4.  46  x|  32  X^,  ^X56 

5.24xl,45x4fx54 

3.  To  multiply  two  fractions  together. 

2  5^2X5^10 

3  7    3X7    21 

Rule.  The  product  of  two  fractions  is  a  new  fraction 
whose  numerator  is  the  product  of  the  numerators  of  the 
two  fractions  and  whose  denominator  is  the  product  of  the 
denominators  of  the  two  fractions. 

In  practice  do  not  actually  carry  out  the  multiplications 
until  you  have  made  as  much  use  as  possible  of  the  principle 
of  dividing  both  terms  by  the  same  number.    Thus: 

36     10 
35^27 

7-6-^3 --9-    21 


4  BEGINNERS'  ALGEBRA 

EXERCISES 

Multiply: 

1   2^3  2    3  5    3  12    15  9     12  45    18 

3  ^7'  3  ^4'  7  ^8  5  ^  4  '  8  ^27'  32'    25 

3    8     30  50    34    8  19         ,    ^^3    15    14  7    6 
^'  15^64'  17^50'  19^7         *•  5^7^'2  ^8l*  6^7 

,    9    20  21     10  .    3     10    21     10  5    2    21 

5^36'  35^"6  6^15^35^14'  7^3^10 

4.  Reciprocal  of  a  number. 

J  is  the  reciprocal  of  2. 
J  is  the  reciprocal  of  3. 

2  is  the  reciprocal  of  J. 

3  is  the  reciprocal  of  |. 
5  is  the  reciprocal  of  -J. 
f  is  the  reciprocal  of  |. 
What  is  the  reciprocal  of  J? 
What  is  the  reciprocal  of  f? 
What  is  the  reciprocal  of  5? 
What  is  the  reciprocal  of  :|-? 
What  is  the  reciprocal  of  4? 

6.  To  divide  a  fraction. 

2^7  =  2     l^A 
3  •        3'^7     21 

2^5^2     7^14 

3*7~3'^5     15 

To  divide  by  a  number  is  the  same  as  to  multiply  by  the 
reciprocal  of  that  number. 

Rule.    To  divide  by  a  fraction  is  the  same  as  to  multiply 
by  the  reciprocal  of  the  fraction. 

EXERCISES 

Divide: 

i7.^6.„15.„  ^7.21.26.3 


INTRODUCTORY  .         5 

„    3  .  7  21  .  9  63^  7  .    15_^  .  3^-  25_^j. 

'^'  4^9'  16  •  8'  25  •  15  **  16       '  8  '  ^'  30 

.72  6_6  8^9  f,    7^^4  49^otr  l^o 

^-  3  *  3'  5  •  3'  9  •  8  ''•  12  •  7'  75  •      '  16  ' 

6.  To  add  two  fractions. 

(a)  If  their  denominators  are  the  same: 

2    5^7 
3"^3     3 

Rule.  Add  the  numerators  for  the  numerator  of  the 
sum  and  use  the  given  denominator. 

(b)  If  the  denominators  are  not  the  same : 

2    5^14     15^29 
3"^7     21  "^21     21 

Rule.  Reduce  to  fractions  having  the  same  denomi- 
nator and  then  add  as  in  (a). 

EXERCISES 

Add: 

1   l+i  2    5  7    2  2    5  3    2  35 

2^3'  3^5'  9^5  3^6'  2^3'  4^6 

3  1.1  §4.1  1.3         4  1+1+1  UM 

10^15'  7^14'  16^4  '  8^32^16'  5^6^3 

^    1_L.2,3,4   1,2,8  f,    3,4,^,_5    4,5,1 

^'  2'^3"^4"^5'  2"^3"^9  ^'  7"^5"^20"^14'  5"^6+l2 

7.  To  subtract  one  fraction  from  another.  Follow  the 
rule  for  adding  fractions,  but  subtract  when  the  rule  says 
add. 


EXERCISES 


Subtract: 


1    1_1  1_1  2__1  2    ?_?  ?-§  1-1 

2    3'  7    9'  3    5  *  7    5'  4    5'  9    12 

q   1_?  1_?   ll-?  4   4_3   1_2  7_3 

12    8'  16    5'   5     2  ■  5    4'  2    5'  6    4 


6  BEGINNERS'  ALGEBRA 

8.  Operations    with    mixed    numbers.    The    following 
typical  illustrations  offer  suggestions: 
(a)  Reduce  a  fraction  to  a  mixed  number : 


33     .  ,5       ^, 

y  =  4+^or4f 

(6)  Reduce  a  mixed  number  to  a  fraction: 

35    3    35+3    38 
^^     5"^5        5         5 

(c)  Add  mixed  numbers: 

2|+3i  =  5f  =  6i 

4J+3J-7f 

(d)  Multiply  mixed  numbers: 

32 
3i 
96        32X3 
16        32Xi 
102 

23 

34 

69 
7f 
76f 

In  multiplying,  one  might,  if  he  chose,  reduce  the  mixed 
nimibers  to  fractions  and  then  multiply: 

23X3i  =  23Xy=^  =  76f 

The  following  is  generally  the  more  convenient  method  in 
case  both  fractions  are  mixed  numbers: 


75X23-2  ^^"2 


EXERCISES 

.    T.   1       .       .J         u  9  81  43  75  17 

1.  Reduce  to  mixed  numbers:     r'  'Wf  To'  00^  'o^' 

2.  Reduce  to  pure  fractions:    3|,  7|,  8f ,  Of,  23f . 

3.  Copy  the  table  at  the  top  of  the  following  page  and  fill  in  by 
adding  each  number  in  the  horizontal  row  to  each  number  in  the 
vertical  column. 


INTRODUCTORY 


2i 

2i 

f 

2| 

4* 

5 

2i 

2i 

5| 

6J 

1? 

3 

4.  Copy  and  fill  in  the  blanks  of  Exercise  3  by  subtracting  each 
number  in  the  horizontal  row  from  each  number  in  the  vertical 
column  whenever  possible. 

5.  Copy  and  fill  in  the  blanks  of  Exercise  3  by  multiplication. 

6.  Copy  and  fill  in  the  blanks  of  Exercise  3  by  dividing  the 
nimibers  in  the  vertical  column  by  those  in  the  horizontal  rows. 

7.  Multiply:    349  X2i,  743  X23f ,  936  Xl4f ,  349  xSh 

9.  Ratio.  When  one  quantity  is  twice  another  quantity, 
the  two  quantities  are  said  to  be  in  the  ratio  2  to  1.  If  one 
quantity  is  1§  times  another,  they  are  said  to  be  in  the 
ratio  3  to  2. 

What  is  the  ratio  of  8  to  4? 
What  is  the  ratio  of  15  to  3  ? 
What  is  the  ratio  of  6  to  2? 
How  many  times  larger  is  9  than  6? 
What  is  the  ratio  of  9  to  6  ? 
What  is  the  ratio  of  3  to  4? 

The  ratio  of  two  numbers  is  their  quotient.  To  find  the 
ratio  of  two  quantities  such  as  3  pints  and  2  quarts,  it  is 


8  BEGINNERS'  ALGEBRA 

necessary  that  these  quantities  be  expressed  in  the  same 
unit. 

EXERCISES 

1.  Find  the  ratio  of  36  to  12. 

2.  Find  the  ratio  of  12  to  15. 

3.  Find  the  ratio  of  3  feet  to  2  feet. 

4.  Find  the  ratio  of  3  inches  to  1  foot. 

5.  Is  the  ratio  of  14  feet  to  2  feet,  7  feet? 

6.  What  is  the  ratio  of  4  to  12? 

7.  What  is  the  ratio  of  |  to  |? 

8.  What  is  the  ratio  of  f  to  f  ? 

9.  Express  the  ratio  3  to  12  in  a  decimal  form. 

10.  Find  the  ratio  of  2  to  7  correct  to  three  decimal  places. 

11.  What  is  the  ratio  of  a  mile  to  a  kilometer?  1  mile  =  1 .  6093 
kilometers. 

12.  The  population  of  the  United  States  was  91,972,266  in  1910 
and  105,610,720  in  1920.  Find  the  ratio  of  the  population  in  1920 
to  that  in  1910  correct  to  two  decimal  places. 

13.  Two  rectangles  have  the  dimensions  6  by  15  and  10  by  18. 
What  is  the  ratio  of  their  areas? 

14.  The  dimensions  of  one  box  are  4  by  6  by  10;  the  dimensions 
of  another  are  6  by  8  by  12.  What  is  the  ratio  of  the  volume  of 
the  first  to  the  volimie  of  the  second? 

15.  If  the  ratio  of  two  numbers  is  3,  the  first  is  how  many  times 
the  second? 

16.  If  the  ratio  of  two  numbers  is  |^,  the  first  is  how  many  times 
the  second? 

10.  Decimals.  Fractions  with  10,  100,  1000,  etc.,  for 
denominators  are  more  conveniently  written  and  used  in  the 
decimal  form.     Thus: 

^-  75    5^-3  4 


INTRODUCTORY  9 

11.  Addition  and  subtraction.  To  add  decimal  numbers, 
place  the  numbers  with  the  units  digits  in  the  same  column 
and  add  as  in  the  case  of  integers.  The  decimal  point  of 
the  result  will  fall  directly  tmder  the  decimal  point  of  the 
ntimbers  to  be  added. 


To  subtract  decimal  nimibers,  follow  the  same  form  as  in 
adding,  subtracting  instead  of  adding. 

Add                                            Subtract 

3.45 
14.3 
7.6 

13.45 
7.63 

25.35 

5.82 

Add: 

1.  34.26,32.5,  1.72 

3.  .036,72.5,89.2 

EXERCISES 

2.   .432, 
4.  96.3, 

7.64,32.55 
493.67,32.4 

Subtract: 

5.  96.37  from  239.42 
7.  2.346  from  32.49 
9.  40.325  from  40.990 

6. 

8. 

10. 

3.75  from  432.7 
362.975  from  762.37 
937.8  from  1000 

12.  Multiplication  and  division  by  10,   100,   1000,  etc. 

To  multiply  by  10,   100,   1000,  simply  move  the  decimal 
point  the  proper  number  of  places  to  the  right,  as  follows: 
2 .  356  X 10  =  23 .  56,  one  place  to  right 
'     2 .  356  X 100  =  235 . 6,  two  places  to  right 
2.356X1000=      ?     ,     ?    places  to  right 
To  multiply  by  200,  move  the  decimal  point  two  places 
to  the  right  and  multiply  by  2 : 

2.314X200  =  231.4X2  =  462.8 
To  divide  by  10,  100,  1000,  etc.,  move  the  decimal  point 
the  proper  number  of  places  to  the  left : 

247 . 3  -^  10  =  24 .  73,  one  place  to  left 
247 . 3  ^  100  =  2 .  473,  two  places  to  left 
2345 . 6  -^  10,000  =    ?,  ?  places  to  left 
2 


10 


BEGINNERS'  ALGEBRA 


To  divide  by  200,  move  the  decimal  point  two  places  to 
the  left  and  divide  by  2. 

To  multiply  by  .01  is  the  same  as  to  multiply  by  ywu*  is 
the  same  as  dividing  by  100;  therefore  move  the  decimal 
point  two  places  to  the  left. 

EXERCISES 

1.  Copy  the  following  table  and  multiply  each  number  in  the 
horizontal  row  by  each  number  in  the  vertical  column  and  place 
the  results  in  the  proper  places: 


32 

24.2 

3.456 

.237 

10 

100 

3000 

' 

.1 

.02 

2.  Copy  the  foUpwing  table  and  divide  each  number  in  the 
horizontal  row  by  each  number  in  the  vertical  column  and  place 
the  results  in  the  proper  places: 


3.42 

47.34 

.354 

906 

10 

1000 

200 

300 

INTRODUCTORY  11 

13.  Multiplication.  Rule.  Multiply  as  if  with  integers 
and  point  off  as  many  decimal  places  in  the  product  as  there 
are  in  both  numbers  that  are  to  be  multiplied  together. 

32.7 
2.43 
981 
1308 
654 
79.461 

The  following  rule  gives  a  better  method  of  multiplying 
decimal  numbers : 

Rule.  Write  down  the  numbers  as  in  addition  with  the 
units  digit  in  the  same  column — that  is,  with  the  decimal 
points  under  each  other — and  then  begin  to  multiply  at 
the  left  of  the  multiplier. 

(a)         32.7  (6) 


32.7 
2.43 

65.4 
13.08 
.981 

32.7 
25.3 

654 
163  5 
9  81 

79.461  827.31 

It  will  be  noticed  that  the  decimal  point  of  the  product 
falls  directly  imder  the  decimal  point  of  the  number  to  be 
multiplied.  The  decimal  points  of  the  partial  products  are 
shown  in  (a),  but  are  omitted  in  the  other  illustrations.  It 
is  important  to  determine  where  to  place  the  first  partial 
product.  In  (a)  we  find  the  first  partial  product  by  multi- 
plying by  2  .The  position  of  the  decimal  point  is  unchanged, 
and  the  last  figure  of  the  partial  product  comes  imder  the 
last  figiu-e  of  the  upper  factor.  The  other  partial  products 
follow  on  just  as  in  ordinary  multiplication. 

In  (b)  we  first  multiply  by  20,  that  is,  we  move  the  decimal 
point  one  place  to  the  right  and  multiply  by  2,  getting  654; 
consequently  4  must  come  in  the  units  column,  that  is,  one 
place  to  the  left  of  the  last  figure  of  the  multiplicand. 


12  BEGINNERS'  ALGEBRA 

(c)  32.7  (d)  32.7 

.32  .032 


9  81  981 

654  654 


10.464  1.0464 

The  student  can  now  readily  see  how  to  find  the  position 
of  the  first  partial  product  in  any  case.  When  that  is  placed, 
the  others  follow  as  in  ordinarv  multiplication. 

EXERCISES 

Multiply: 

1.2.3X7.2  2.14X3.5  3.  14  X. 72 

4.  4.2X6.71  5.  24x7.42  6.  5^X83.7 

7.  24.7X39.2  8.  3.52x69.3  9.  7^x36.26 

10.  .36X.0362  11.  6.34X.052  12.  83.25X.6321 

14.  Division.  Probably  the  best  way  of  dividing  by  a 
decimal  is  to  reduce  the  divisor  to  an  integer  by  multiplying 
by  its  decimal  denominator.  Of  course  the  dividend  must 
then  be  multiplied  by  the  same  number.  This  amounts 
to  moving  the  decimal  point  the  same  ntunber  of  places  to 
the  right  in  both  divisor  and  dividend.  The  quotient  is  to 
be  placed  above  the  dividend;  the  first  figtu^e  of  the  quotient 
above  the  right-hand  figure  of  the  first  partial  dividend. 
Thus: 

(a)     43.27^3.2  (6)      3.764-M2.37 

13.5  .304 

32|432.7  1237|376.4 

32  371 1 


112  5  300 

96_ 
167 
160 

7 
The  decimal  point  of  the  quotient  falls  above  the  decimal 
point  of  the  dividend. 


INTRODUCTORY  13 

EXERCISES 

Divide: 

1.  .73-5-9  2.36.2-^9.3  3.5.73-^.6 

4.423-^.73  5.  .025^.73  6.45.2-^-423 

7.  .73-1-423  8.  45.2-^•.73  9.  9.3^.025 

10.3.52-69.3  11.  .0036^.092  12.  .035-^.0023 

15.  Use  of  approximate  nixmbers.  Find  the  circumference 
of  a  circle  of  diameter  27  inches.  You  have  been  taught  to 
compute  the  circumference  by  multipl)mig  the  diameter  by 
•^  =  3^  or  by  3. 1416. 

Using  each  of  these  values,  we  find : 

ciramif  erence  =  27  X  3|  =  84f  =  84 .  857 
circumference  =  27  X  3 .  1416  =  84 .  8232 

These  results  agree  in  the  first  three  figures  on  the  left, 
but  differ  in  the  other  figures.  Both  are  correct  as  far  as 
they  agree ;  neither  is  correct  beyond  that  point.  The  reason 
for  this  is  that  the  number  used  as  a  multiplier  of  the  diameter 
is  not  accurately  known.  The  values  3^  and  3.1416  are 
merely  approximations  to  the  number  that  should  be  used. 
The  nimiber  that  should  be  used  is  denoted  by  the  Greek 
letter  tt  (pronounced  pi) .  It  has  been  shown  that  w  cannot 
be  exactly  expressed  by  a  fraction  or  a  decimal.  Out  to  ten 
places  the  number  is  3 .  1415926536. 

3 .  1416  is  a  five-figure  approximation. 

3 .  142  is  a  four-figure  approximation. 

■^  or  34"  is  a  fractional  approximation  that  has  been  known 
for  two  thousand  years  or  more.  It  is  not  as  accurate  as 
either  of  the  other  two,  as  is  shown  when  it  is  reduced  to  the 
decimal  form: 

3^  =  3.14287+ 

This  value  3-f  is  somewhat  easier  to  use  and  gives  results 
close  enough  in  ordinary  cases  where  only  three  figiu*es  are 
used. 


14  BEGINNERS'  ALGEBRA 

Comparing  results,  we  have 

27X31  =  84.857 
27X3.142  =  84.834 
27X3.1416  =  84.8232 

The  last  is  the  most  accurate.  A  still  more  accurate 
result  would  be  obtained  by  using  a  more  accurate  approxi- 
mation for  TT,  as,  3.14159: 

27X3.14159  =  84.82293 

You  will  notice,  in  the  comparison  above,  that  each  result 
has  figures  at  the  right  that  are  shown  to  be  incorrect  when 
closer  approximations  are  made.  Such  figures  are  useless. 
This  fact  leads  to  the  very  important  rule  that  should 
always  be  observed  when  calculations  with  approximate 
numbers  are  being  made: 

Rule.  A  result  of  a  calculation  with  approximate  num- 
bers should  not  show  more  figures  than  do  the  approximate 
numbers  used. 

For  instance,  in  the  calculations  given  above,  the  product 
27X3.1416  should  not  show  more  than  five  figures;  the 
product  27X3. 142  should  not  show  more  than  ioux  figures; 
the  result  of  the  multiplication  should  be  cut  back  to  the 
proper  number  of  figures. 

For  discarding  the  meaningless  figures  at  the  right  of  such 
calculated  results  the  common  practice  is  indicated  in  the 
following  rule : 

Rule.  Determine  how  many  figures  you  wish  to  retain, 
then,  if  the  next  figure  to  the  right  is  less  than  5,  throw  away 
all  figures  not  to  be  retained ;  if  the  next  figure  to  the  right 
is  5  or  greater  than  5,  add  one  to  the  last  digit  retained  and 
throw  away  all  figures  not  to  be  retained. 

For  instance; 

27X3.14  =  84.718 
Retain  three  figures,  result  84.8 


# 

INTRODUCTORY  15 

27X3.142  =  84.8314 
Retain  four  figures,  result  84.83 

27X3. 1416  =  84. 823|2 
Retain  five  figures,  result  84 .  823 

If  34"  is  used  for  tt,  only  three  figures  should  be  retained 
because  34- =  3. 1429  is  itself  accurate  to  only  three  places; 
hence : 

27X3i  =  84.857+ 

Retain  three  figures,  result  84 . 9 

Remember  whenever  34-  is  used  for  tt  never  to  give  a  result 
showing  more  than  three  figures.  For  instance,  find  the 
circumference  of  a  circle  526  feet  in  diameter: 

526X31  =  1653.1 

Retain  three  figures,  result  1650 

When  the  3  is  thrown  away,  its  place  must  be  filled  by  0 

so  that  the  decimal  point  shall  come  in  the  right  place.     It 

must  be  remembered  that  in  all  such  cut-backs  the  last 

figure  retained  is  doubtful,  being  either  too  large  or  too  small. 

16.  Numbers  obtained  by  measuring.  There  is  still 
another  matter  to  be  taken  into  con- 
sideration. The  machinist  uses  cali- 
pers (Fig.  1)  for  measuring  directly 
the  diameter  of  a  pipe,  shaft,  or  cir- 
cular rod. 

Suppose  the  diameter  of  a  pipe  has 
been  measured  to  the  nearest  tenth  of  i  •    -      a  tpers 

an  inch  and  found  to  be  2 . 7  inches.     Using  tt  =  3 .  1416, 
circumference  =  2. 7X3. 1416  =  8.4823 

When  the  diameter  was  measured  more  accurately,  say 
to  hundredths  of  an  inch,  it  was  found  to  be  2 .  73  inches. 
Now, 

circumference  =  2. 73  X3. 1416  =  8. 5766 


16  BEGINNERS'  ALGEBRA 

The  last  restilt  is  the  more  accurate  because  a  more  accu- 
rate measurement  of  the  diameter  is  used. 

The  measure  of  the  diameter  is  itself  an  approximate 
nimiber.  This  leads  to  a  slight  amendment  to  the  rule,  as 
follows : 

Rule.  Do  not  retain  more  figures  in  the  result  of  a 
multiplication  of  approximate  numbers  than  there  are  in 
that  one  of  the  approximate  numbers  used  which  has  the 
least  number  of  figures. 

In  the  multiplication 

2.73X3.1416  =  8.5766 
since  2.73  has  but  three  figures,  the  result  should  be  cut 
back  to  three  figures,  namely  8.58.     Had  3|  been  used 
instead  of  3 .  1416,  the  same  result  would  have  been  reached, 
8,58. 

To  give  another  illustration:  The  edges  of  a  rectangular 
card  when  measured  to  the  nearest  tenth  of  an  inch  are  found 
to  be  6.7  and  3.8.  In  the  calculation  the  doubtful  figures 
are  in  black  type.  Only  two  figures  are  to  be  retained  in 
the  result. 

6.7 
3.8 


201 
5  36 


25.46    Result,  25 

When  more  accurate  measurements  were  made,  the 
edges  were  found  to  be  6 .  76  and  3 .  82.  The  calculation  is 
as  follows: 

6.76 
3.82 
20  28 
5  408 
1352 
25.8232    Result,  25.8 


INTRODUCTORY  17 

17.  Division  of  approximate  numbers.  The  rule  just 
given  for  determining  the  proper  number  of  figiures  to  show 
in  the  result  of.  a  multipHcation  applies  also  to  division. 

EXERCISES 

The  problems  given  below  are  supposed  to  be  practical;  the 
numbers  given  are  the  results  of  measurement  and  are  therefore 
approximate,  the  right-hand  figiue  being  in  doubt. 

1.  Find  the  circumference  of  a  circle  of  diameter  1.86  inches; 
use  3i,  3 .  142,  and  3. 1416  and  compare  results. 

2.  Find  the  circumference  of  a  circle  of  diameter  5.7;  use  S^  and 
3 .  142  and  compare  results. 

3.  A  tree  measures  25|-  feet  arotmd  its  trunk.  What  is  its 
diameter? 

4.  The  area  of  a  circle  is  found  by  multiplying  the  square  of  the 
radius  by  tt.    Find  the  area  of  a  circle  of  radius  4 . 27  inches. 

5.  A  5-foot  concrete  walk  is  laid  outside  a  circular  grass  plot 
60  feet  in  diameter.    What  will  it  cost  at  9  cents  a  square  foot? 

6.  Find  the  diameter  of  a  circle  whose  circumference  is  43.62 
inches. 

7.  Find  the  diameter  of  a  stove  pipe,  steam  pipe,  water  pipe,  or 
any  convenient  circular  pipe  or  tank  by  measuring  the  circum- 
ference. 

8.  Find  the  diameter  of  a  nimiber  of  trees  in  your  vicinity.  Is 
there  any  other  way  of  finding  the  diameter  of  a  tree? 

Note,  The  forester  uses  an  instrument  called  tree-calipers  for 
finding  the  diameter  of  a  tree. 

9.  Measure  the  diameter  of  a  pipe  with  calipers;  also  measure 
the  circumference  of  the  same  pipe  with  a  tape.  To  test  the 
accuracy  of  the  measurements  compute  the  circumference  and  com- 
pare with  the  direct  measure.     Explain  the  resulting  situation. 

10.  A  certain  rectangular  lot  was  found  to  be  34.62  feet  by 
74.08  feet.    What  is  its  area? 

11.  The  inside  dimensions  of  a  box  are  carefully  measured  and 
found  to  be  3.55,  6.24,  and  5.17  inches.      What  is  its  capacity? 


18  BEGINNERS'  ALGEBRA 

12.  If  the  dimensions  of  the  bottom  of  a  box  are  2  feet  6  inches 
and  4  feet  3  inches,  how  deep  should  the  box  be  that  it  may  contain 
1  cubic  yard? 

13.  If  1  cubic  foot  of  water  weighs  63.255  pounds,  what  does  a 
quart  of  water  weigh?    57 .  75  cubic  inches  is  a  Uquid  quart. 

14.  A  dry  quart  is  equivalent  to  67.20  cubic  inches.  Does  the 
ordinary  square  strawberry  box  contain  a  quart  when  level  full? 

18.  Order  of  operations.  As  far  as  the  result  is  concerned, 
it  does  not  matter  in  what  order  several  additions  are  made : 

3+7+5  =  10+5  =  15 
or  =3+12  =  15 

or  =8+7  =  15 

Sometimes  the  result  is  obtained  more  quickly  if  the  addi- 
tions are  grouped  in  tens: 


I      1 i       I 

5+9+4+7+1+3+6  =  35 

I ^1 

5+10+10+10  =  35 

So  also  in  a  sum  of  additions  and  subtractions,  the  order  in 
which  the  work  is  done  does  not  matter. 

.     7-2+5=7+5-2  , 
=  12-2 
=  10 

We  usually  proceed  from  left  to  right  when  possible,  thus : 

23-4+3-7  =  19+3-7 
=  22-7 
=  15 

But  in  7+2-12+8  =  9-12+8, 

as  we  cannot  subtract  12  from  9,  we  must  add  9+8  first  and 
get 

17-12  =  5 

In  a  series  of  additions,  subtractions,  multiplications,  and 


INTRODUCTORY 


19 


divisions  the  multiplications  and  divisions  are  to  be  done 
before  the  other  operations,  thus: 

2+3X5-7+6-^3  =  2+15-7+2 
=  17-7+2 
=  10+2 
=  12 
The  order  in  which  the  operations  are  to  be  done  may 
be  changed  by  the  use  of  parentheses,  thus  • 

(2+3)X5-7+6-f-3  =  5X5-7+6^3 
=  25-7+2 
=  18+2 
=  20 
The  operations  within  the  parentheses  must  be  done  first 

EXERCISES 


1.  25+13+15 
3.  8+7+2+3 
5.  18-7+3 
7.  9x6+3 
9.  9x(6+3) 


2.  7+19+3 
4.  4+9+1+6+8 
6.  18 -(7+3) 
8.  9+6x3 
10.  18-9+3-7 


11.  9-13+15 


12.  |X6+2 


13.  |x(6+2) 

15.  2x(7-3)+2-7 


17. 


5+2    7-3 


8 


16 


14.  3x6+5-7+3x2 


16.  5X 


7+2 


CHAPTER  II 
Formulas  and  Equations 

19.  The  formula.  How  many  square  feet  are  there  in 
the  floor  of  a  room  14  feet  by  18  feet?  How  many  square 
yards  are  there  in  a  tennis  court  36  feet  by  78  feet?  What 
is  the  area  of  a  city  lot  40  feet  wide  and  150  feet  deep? 
How  would  you  find  the  area  of  a  card  having  square  comers? 

In  words  the  answer  is:  The  area  of  a  rectangle  is  found 
by  multiplying  its  length  by  its  width. 

This  rule  may  be  stated  more  briefly  by  the  omission  of 
unnecessary  words  and  by  the  use  of  certain  symbols: 
area = length  X  width 

This  may  be  still  further  shortened  by  the  use  of  the  first 
letter  of  each  word:  a  =  lXw 

We  may  even  omit  the  times  sign  and  agree  that  the  two 
letters  Iw  standing  together  shall  mean  "length  times 
width"  and  write  the  rule  in  the  form 

a  =  lw 

The  multiplication  sign  cannot  be  omitted  when  two 
arithmetical  numbers  are  used  such  as: 

2X5,  for  25  already  has  another  meaning 

2  2 

2Xk,  for  2f  already  means  2+^ 
o  o 

It  is  convenient  to  have  a  special  name  for  such  a  short- 
hand way  of  writing  a  rule.  The  Latin  word  formula  is  an 
appropriate  word  to  use,  for  it  means  "short  form." 

It  has  become  a  very  common  practice  to  write  rules  in 
this  shorthand  fashion.  Such  a  formula  is  easy  to  remember, 
its  meaning  is  quickly  grasped,  and  it  is  easily  used  in  any 

20 


FORMULAS  AND  EQUATIONS        ,  21 

particular  example.     The  use  of  formulas  leads  to  orderly 
ways  of  setting  down  work. 

Illustration.    How  many  square  feet  are  there  in  a  5-foot  concrete 
walk  in  front  of  a  40-foot  lot? 
Rule  or  formula  to  be  used: 

a=lw 
Here  /  is  40  and  w  is  5 

Hence  we  have  a  =  40  X  5 

=  200 

EXERCISES 

Work  the  following  examples  in  the  same  way: 

1.  Find  the  area  of  a  window  glass  13  inches  by  15  inches. 

2.  Find  the  area  of  a  wall  9  feet  by  22  feet. 

Express  the  following  ndes  as  formulas,  using  the  first  letters  of 
the  quantities: 

3.  Rule  for  finding  the  total  cost  of  a  number  of  articles  when  the 
price  of  one  axticle  is  given. 

4.  Rule  for  finding  the  number  of  inches  in  a  given  number  of  feet. 
6.  Rule  for  changing  pounds  into  ounces. 

6.  Rule  for  computing  the  circumference  of  a  circle  if  the  diameter 
is  given. 

7.  Any  other  rules  that  you  can  call  to  mind  in  which  one  number 
is  found  by  multiplying  together  two  other  numbers. 

8.  Rule  for  finding  the  volume  of  a  room. 

9.  Rule  for  finding  the  area  of  a  triangle. 

10.  Make  numerical  examples  to  fit  each  of  the  formulas  of  Exer- 
cises 3-9. 

11.  What  must  be  the  length  of  a  board  9  inches  wide  if  its  area 
is  to  be  144  square  inches? 

The  rule  for  finding  the  length  of  a  rectangle  when  the  area  and 
width  are  given  is 

length = area  divided  by  width 

or,  better,  length  =  -vr-r 

^        width 

What  is  the  shorthand  form?    Find  the  answer  to  the  exercise 


22  BEGINNERS'  ALGEBRA 

12.  Write  in  shorthand  the  rules  for  finding: 

a)  The  price  of  one  article  when  the  cost  of  a  niimber  is  given. 

b)  The  diameter  of  a  circle  when  the  circumference  is  given. 

c)  The  number  of  dozens  in  a  given  nimiber  of  eggs. 

d)  The  number  of  yards  in  a  given  nimiber  of  feet. 

13.  Make  numerical  exercises  to  fit  each  of  the  formulas  in  Exer- 
cise 12. 

14.  Find  the  number  of  inches  in  3  feet  5  inches. 

15.  Write  the  formula  for: 

a)  The  number  of  inches  in  a  measurement  given  in  feet  and 

inches. 

b)  The  number  of  pints  in  a  measurement  given  in  quarts  and 

pints. 

c)  Computing  the  year  in  which  you  were  born  from  your  age. 

d)  Rule  for  making  coffee,  one  teaspoonful  for  each  person 

and  one  for  the  pot. 

16.  Make  numerical  examples  for  each  of  the  formulas  in  Exer- 
cise 15. 

17.  Write  a  formula  for  finding  the  perimeter  of  a  rectangle. 

20.  The  formula  states  a  relation.    The  formula 

a  =  lw 

is  a  short  way  of  writing  the  rule  for  finding  the  area  of  a 
rectangle  when  the  length  and  width  are  given. 

The  formula  Z  =  - 

w 

is  a  short  way  of  writing  the  rule  for  finding  the  length  when 
the  area  and  width  are  given. 

The  important  thing  for  us  to  notice  is  that  the  formula 

a  =  lw 

really  includes  the  formula    I  =  — 

One  involves  multiplication  and  the  other  division. 


FORMULAS  AND  EQUATIONS  23 

One  asks  the  question       ?  =  4  X  5 

20 
the  other  asks  the  question  ?  =  -f" 

or,  in  another  form,  5  X  ?  =  20 

What  question  does  a  =  7  X  8  ask  ?  How  answered  ? 
What  question  does  /  X  7  =  28  ask  ?  How  answered  ? 
What  does  9ze;  =  26  ask? 

The  formula  a  =  lw  is  something  more  than  a  rule:  it  is  a 
statement  of  the  fact  that  the  length,  width,  and  area  of  a 
rectangle  are  so  related  to  one  another  that  the  area  equals 
the  product  of  the  length  and  the  width.  This  may  be 
called  the  area  law  of  the  rectangle.  If  any  two  of  these 
three  numbers  are  known,  the  other  one  can  be  determined. 
We  then  really  have  need  of  only  the  one  formula  a  =  lw 
for  this  class  of  problems,  or  lw  =  a,  for  it  matters  not  which 
way  the  equality  is  written. 

Illustration.  How  wide  is  a  room  if  its  length  is  22  feet  and  its 
area  is  396  square  feet? 

Formula  lw=a  where  /  =  22,  a =396 
22w=396 

396 

w= — • 
22 

=  18 

PROBLEMS 

In  solving  the  following  nine  problems  write  a  formula  for  each 
and  then  solve  the  problem: 

1.  A  farm  of  60  acres  sold  for  $4,500.  What  was  the  price  per 
acre? 

2.  A  city  lot  sells  for  $5000,  the  rate  being  $125  a  front  foot. 
Find  the  number  of  front  feet  in  the  lot. 

3.  A  man  who  eats  his  dinners  at  restaurants  finds  that  for  a  cer- 
tain week  his  dinners  cost  him  $4 .  50.  What  was  the  average  cost 
of  a  dinner? 


24  BEGINNERS'  ALGEBRA 

4.  A  certain  grade  of  pencils  sells  at  wholesale  for  $12.96  a  gross. 
What  is  the  cost  per  pencil? 

5.  How  much  water  flows  through  a  pipe  in  one  minute  if  450 
gallons  are  delivered  in  2^  hours? 

6.  The  dimensions  of  a  box  are  9^,  3^,  and  16  inches.  How 
many  cubic  inches  does  the  box  contain? 

7.  What  is  the  height  of  a  box  8  inches  wide  and  18|-  inches  long 
if  it  contains  988  cubic  inches? 

8.  What  must  be  the  depth  of  a  box  6  inches  in  length  and  3^ 
inches  in  width  to  contain  a  pint  (231  cu.  in.  =  l  gal.)? 

9.  A  printers'  rule  for  computing  the  charges  to  be  made  on  a 
certain  kind  of  circular  is  75  cents  for  setting  up  and  25  cents  a 
himdred  for  printing.    What  was  the  bill  for  700  circulars? 

Solve  the  following  six  problems  by  use  of  but  one  formula: 

10.  What  is  5  per  cent  of  75? 

11.  21  is  what  per  cent  of  350? 

12.  24  is  8  per  cent  of  what  number? 

13.  35  is  what  per  cent  of  210? 

14.  36  is  5  per  cent  of  what  nimiber? 

15.  42  is  6  per  cent  of  what  number? 

16.  Write  formulas'  for  the  following  rule:  To  find  gain  or  loss 
multiply  the  cost  by  the  rate  per  cent  of  gain  or  loss.  Use  c  for 
cost,  r  for  rate,  g  for  gain,  I  for  loss. 

17.  What  does  a  man  gain  on  goods  that  cost  $720  if  he  sells 
them  12^  per  cent  above  cost? 

18.  If  goods  cost  $480  and  are  sold  at  a  gain  of  $80,  what  is  the 
gain  per  cent? 

19.  What  is  the  cost  of  goods  if  $96  is  gained  when  they  are  sold 
8^  per  cent  above  cost? 

20.  What  does  a  man  lose  on  goods  that  cost  $360  if  he  sells  them 
at  6  per  cent  below  cost? 

21.  What  is  the  cost  of  goods  if  $20  is  lost  when  they  are  sold  at 
5  per  cent  below  cost? 


FORMULAS  AND   EQUATIONS  25 

22.  What  connection  is  there  between  Problems  16-21  and 
Problems  10-15? 

23.  From  the  formula  irD=15^  find  D.  What  is  the  meaning  of 
the  formula? 

24.  If  a  train  travels  at  the  average  rate  of  40  miles  an  hour,  how 
far  does  it  go  in  2  hours?  In  3  hours?  In  f  hour?  In  i  hour?  In 
10  minutes?    In  /  hours? 

State  as  a  fomiula  the  rule  jiist  used,  putting  d  for  distance,  r  for 
rate,  and  /  for  time.  Use  this  formula  in  working  the  following  five 
problems. 

25.  How  far  can  an  airplane  fly  in  3  hours  at  the  rate  of  120 
miles  an  hour? 

26.  A  train  travels  from  Chicago  to  St.  Louis,  a  distance  of 
294  miles,  in  8  hours.  What  is  the  average  speed  per  hour,  not 
counting  stops? 

27.  How  long  will  it  take  an  automobile  going  at  the  rate  of  15 
miles  an  hour  to  travel  35  miles? 

28.  How  long  will  it  take  the  automobile  mentioned  in  Problem  27 
to  go  tjie  length  of  a  city  block  of  300  feet? 

29.  It  was  noticed  that  an  automobile  traveled  a  block  300  feet 
long  in  10  seconds.  Was  it  exceeding  the  speed  limit  of  15  miles 
an  hour? 

Three  times  a  niimber  added  to  2  times  the  same  number 
is  30,  may  be  written  in  the  short  form,  n  being  used  for  the 
number : 

3w+2w  =  30 

The  directions  for  finding  the  number  are  not  given. 
Can  you  find  out  what  to  do  in  order  to  find  the  number? 
Ask  yourself  the  question:  How  many  times  a  number  is 
3  times  the  number  and  2  times  the  number? 

Work  out  the  following  four  problems  in  the  same  way: 

30.  Nine  times  a  number  is  35.    What  is  the  number? 
3 


26  BEGINNERS'  ALGEBRA 

31.  Seven  times  a  number  less  4  times  the  number  is  123.  What 
is  the  nimiber? 

32.  Seven  times  a  nimiber  increased  by  4  times  the  nimiber  is 
132.    What  is  the  number? 

33.  What  nimiber  added  to  5  times  itself  equals  102? 

21.  Directions  for  work  not  given.  A  formula  states  a 
rule  for  finding  some  required  number.  For  instance,  when 
the  area  and  width  of  a  rectangle  are  given,  the  length  is 
found  by  use  of  the  formula 

w 
The  directions  for  the  work  are  all  indicated.     We  could 
use  the  f  ormiila 

lw  =  a 
for  solving  the  same  problem.     In  this  case  not  all  the  direc- 
tions for  finding  I  are  given,  and  we  must  think  a  little  to 
find  out  just  what  to  do.     We  must  answer  the  question 

?Xw=a   \ 
The  name  equation  is  used  for  both  types  of  formulas. 

22.  Any  letter  used  for  the  unknown.  In  the  formulas 
thus  far  we  have  been  using  the  first  letters  of  the  quantities 
to  stand  for  the  numerical  values  of  those  quantities.  This 
is  sometimes  very  helpful  and  convenient,  but  it  is  not 
necessary  to  confine  ourselves  to  first  letters.  We  may  use 
any  letter  we  choose  to  represent  the  unknown  in  a  formula 
or  an  equation. 

Five  times  an  unknown  number  plus  2  times  the  same 
number  equals  14. 

5?+2?  =  14 

5w+2m  =  14 

5x+2x  =  14: 

The  three  expressions  all  have  the  same  meaning.    The 

letter  x  has  been  used  so  often  to  represent  the  unknown 


FORMULAS  AND  EQUATIONS  27 

number  that  when  Roentgen  discovered  a  new  ray  which 
was  ahnost  entirely  strange  to  him  he  called  it  the  X-ray, 
the  unknown  ray. 

The  use  of  a  letter  to  stand  for  a  number  whose  value  is 
unknown  is  one  of  the  important  ideas  of  algebra. 

23.  Operations  on  equations.  Let  us  examine  caref\illy 
the  way  in  which  such  an  equation  as  is  given  in  the  last 
article  is  worked  out  to  find  the  tinknown  number. 

The  problem  stated  in  algebraic  language  is 
5w+2n  =  14 

We  are  to  regard  this  equation  as  asking  the  question: 
What  value  can  be  given  to  n  that  will  make  the  statement 
true?  One  sees  at  once  that  5  times  a  number  added  to  2 
times  the  same  number  is  7  times  that  number,  and  we  may 
write  7w  =  14 

The  7  was  foimd  by  adding  together  the  5  and  the  2.     This 
operation  may  be  expressed  on  paper  thus: 
5w+2w  =  (5+2)w 
=  7n 

The  parenthesis  (  )  is  used  to  denote  the  fact  that  5  and  2 
are  to  be  added  together  and  the  result  used  as  a  multiplier 
of  n. 

Now  taking  the  equation 

7w  =  l4 
we  say  that  if  7  times  a  ntimber  equals  14,  the  number  will 
be  ^  of  14. 

This  operation  may  be  written 

w  =  =ofl4 

14 

or,  better,  w=-«- 

=  2 

That  is,  we  may  divide  the  14  by  7. 


28  BEGINNERS'  ALGEBRA 

It  is  to  be  noticed  that  all  the  things  that  were  done  to  find 
the  unknown  number  may  be  written  down  in  algebraic 
shorthand  and  arranged  in  a  clear  and  orderly  form: 

5w+2w  =  14  (1) 

(5+2)w  =  14  (2) 

7n  =  14  (3) 

W  =  y  (4) 

n=  2  (5) 

The  important  thing  to  notice  now  is  the  way  in  which 
we  passed  from  one  equation  to  the  next.  Wc  did  something 
to  each  equation  in  order  to  get  the  next  equation.  To  get 
(2)  from  (1),  we  added  5  and  2;  (3)  is  merely  a  condensed 
form  of  (2).     In  getting  (4)  from  (3)  we  divided  14  by  7. 

We  may,  however,  look  at  this  last  step  in  a  slightly  dif- 
ferent way.  If  we  divide  3X5  by  3,  the  result  is  5.  If  we 
divide  7X6  by  7,  the  result  is  6.  If  we  divide  the  product 
of  any  two  factors  by  one  of  the  factors,  the  result  is  the 
other  factor.  If  we  divide  In  by  7,  the  result  is  n.  We 
may,  then,  consider  that  we  obtain  equation  (4)  from  (3) 
by  dividing  both  sides  by  7.  Equation  (5)  shows  (4)  in  a 
more  condensed  form. 

This  way  of  looking  at  the 
matter  opens  up  a  very  impor- 
tant point  of  view.  The  equa- 
tion states  the  fact  that  two 
numbers  are  equal  to  each 
other.  It  is  to  be  regarded  as 
a  sort  of  balance,  like  a  drug- 
^'^^•^  gist's  pan  balance  (Fig.  2). 

If  any  object  be  placed  in  one  pan,  that  pan  will  go  down 
imless  an  equal  weight  is  put  in  the  other  pan.  The  same 
weight  must  be  in  each  pan  to  keep  the  instrument  in  balance. 


FORMULAS  AND  EQUATIONS  29 

The  two  sides  of  an  equation  are  like  the  two  arms  of 
the  balance  (Fig.  3).     Nothing  must  ever  be  done  to  the 

5n  +  2n  14  In  14 


Fig.  3 

equation  that  will  disturb  this  balance.  If  the  value  of  one 
side  be  changed  in  any  way,  the  other  side  must  be  changed 
in  exactly  the  same  way.  The  problem  at  issue  is  always  to 
find  the  value  of  the  unknown.  The  unknown  niimber  is 
more  or  less  tangled  up  with  the  known  numbers  in  the 
equation.  We  must  do  something  to  the  equation  to  untangle 
the  unknown.  We  have  just  discovered  two  operations  that 
can  be  used  in  untangling  the  imknown  without  disturbing 
the  balance  of  the  equation,  namely: 

Operation  1.     Any  indicated  operation  within  a  side  of 
the  equation  can  be  carried  out. 

Operation  2.    Both  sides  of  an  equation  can  be  divided 
by  the  same  number. 

Illustration .  9» — 5« = 12 

4»=12  (operation  1) 

»  =  3  (operation  2) 

EXERCISES 

Find  the  value  of  the  unknowns  in  the  following  equations, 
indicating  in  each  step  the  operation  used: 

1.  4jc+7jc  =  69  2.  5a4-9a  =  98 

3.  l2x-2x-\-bx=lb  4.  8w-5w+6w-4»=30 

5.  ll^-5/+9^-3/=72  6.  5y+7y-6y-y=16+6-7 

7.  8/-3/+4/-2^=30-54-3  8.  7x+5a;-9a;+2a:=30-12-f2 

Find  value  of: 

9.  Zx-^^x-2x  if  x=b  10.  7a+2-3a-l  if  a=7 

11.  96-7-26+12  if  6=5  12.  8(;+6-5c-3  if  c=4 


30  BEGINNERS'  ALGEBRA 

24.  Definitions.  It  is  convenient  to  have  names  for  things 
to  which  we  constantly  refer.  We  have  already  spoken  of 
the  sides  of  an  equation,  the  parts  on  either  side  of  the 
equality  sign.  The  parts  on  one  side  that  are  separated  by 
addition  or  subtraction  signs  are  called  terms.  In  7w+4w, 
In  is  a  term;  so  also  is  4n. 

A  term  may  have  only  one  ntmiber  in  it  or  it  may  be  made 
up  of  several  factors.  The  part  of  the  term  that  is  repre- 
sented by  an  Arabic  numeral  is  called  the  numerical  coeffi- 
cient of  the  term  or,  more  simply,  the  coefficient  of  the  term. 
In  the  term  7w,  7  is  the  numerical  coefficient.  In  the  term 
Zhw,  3  is  the  ntmierical  coefficient.  What  is  the  coefficient 
of  the  term  lltf     Of  .mpt? 

No  coefficient  appears  in  the  term  n;  in  such  cases  it  is  to 
be  understood  that  the  coefficient  is  1. 

The  letter  used  to  stand  for  a  number  the  actual  value  of 
which  is  not  known  is  called  the  unknown. 

To  solve  an  equation  is  to  find  the  value  of  the  unknown. 
The  actual  value  of  the  unknown  is  called  the  root  of  the 
equation. 

In  the  solving  of  an  equation  certain  operations  are  carried 
out  which  result  in  the  finding  of  a  certain  value  for  the 
imknown.  Mistakes  in  the  work  may  have  been  made.  In 
order  to  discover  if  the  value  found  is  really  the  true  value, 
we  must  check  the  work. 

25.  Checking.  The  pupil  checks  the  solution  of  an 
equation  by  putting  the  value  found  in  place  of  the  letter 
that  stands  for  it  in  the  equation,  and  then  working  down 
each  side  by  itself,  arithmetically.  If  the  two  sides  come 
out  the  same,  it  is  generally  safe  to  say  that  the  work  is 
correct  and  that  the  value  found  is  the  root  of  the  equation. 
If  they  do  not  come  out  the  same,  there  is  trouble  some- 
where and  a  search  should  be  made  for  the  mistake.  Mis- 
takes may  occur  either  in  finding  the  root  or  in  checking 


FORMULAS  AND  EQUATIONS  31 

Illustration,     Take  the  example  solved  in  Art.  23,  page  29, 
9w~5w  =  12 
where  we  found  n  =  S 

Substitute  3  for  w,         9  •  3  -5  •  3  =  12 
27-15  =  12 
12=12 
The  number  3  satisfies  the  equation;  the  work  is  correct. 

EXERCISES 

Solve  and  check: 

1.  6»+3»  =  72  2.  5x-\-8x-9x=4tS 

3.  7/-2/+5/=95  4.  8x+7^-9x  =  70 

5.  9a+Sa-3a  =  70  6.  6a;-2:r+9^-3x  =  72 

7.  36+56-25-6=17-9+2     8.  9c-3c+2c-c=14+10-3 

Find  the  value  of: 

9.  llc-3-2c+7  if  c  =  4  10.  5a+a-8+12-3a  if  a  =  5 

26.  The  subtraction  operation.  Consider  the  problem: 
What  niimber  added  to  7  equals  15?  This  can  be  worked 
very  quickly  by  arithmetic  by  the  simple  device  of  sub- 
tracting 7  from  15.  Translating  the  problem  into  algebraic 
language,  we  have  the  equation 

7+w  =  15 
which  asks  the  question   7+  ?  =  15 
By  arithmetic  we  have  at  once 
w  =  15-7 
=  8 
.    Supply  the  missing  nimiber  in  each  of  the  following : 

?+4  =  7 

M+2  =  9 

3+^  =  11 

12  =  3+? 

15  =  w+7 

17+w  =  17 


32  BEGINNERS'  ALGEBRA 

Now  let  us  consider  the  matter  from  the  algebraic  point 
of  view.     The  statement 

7+w  =  15 
is  an  equation.  We  must  do  nothing  to  it  that  will  destroy 
its  balance.  Since  we  obtained  the  answer  by  subtracting  7 
from  the  right-hand  side,  we  must  also  subtract  (or  take 
away)  7  from  the  left  side  of  the  equation.  We  have  thus 
found  another  operation  that  can  be  applied  to  an  equation, 
namely : 

Operation  3.    The    same   number   can   be    subtracted 
from  both  sides  of  an  equation. 

The  solution  of  the  equation  should  be  written  down  in 
the  following  orderly  form: 

7+w  =  15 

n  =  15-7 

=  8 

Solve  the  equation,     3w + 8  =  20 

The  unknown  nimiber  Sn  is  tangled  up  with  8  by  addition. 
We  may  untangle  it  by  using  the  subtraction  operation  and 

3n  =  12 

Here  n  is  tangled  up  with  3  by  multiplication.     We  may 
untangle  it  by  the  use  of  the  division  operation  and 

w  =  4 

Notice  that  the  end  in  view  was  to  get  the  imknown  niun- 
ber  n  all  alone  on  one  side  of  the  equation  and  at  the  same 
time  have  nothing  on  the  other  side  except  known  numbers. 
To  do  this  it  was  necessary  to  get  rid  of  the  term  that  had 
no  unknown  in  it  and  also  to  get  rid  of  the  multiplier  3.  We 
chose  the  operations  that  would  do  these  things : 
Subtraction  of  a  term  to  get  rid  of  a  term 
Division  by  a  multiplier  to  get  rid  of  a  multiplier 


FORMULAS  AND  EQUATIONS  33 

The  work  should  be  set  down  as  follows: 

3w+8  =  20 
Subtract  8  from  both  sides, 

3m  =  20-8 

3w  =  12 

12 
Divide  both  sides  by  3,  w  =  -^ 

=  4 

EXERCISES 
Solve  and  check: 

1.  a;H-18  =  23  2.  x+3  =  15  3.  19+w  =  25 

4.  15+a;  =  40  5.  27  =  «+17  6.  23  =  3ic+5 

7.  3o+4=19  8.  100  =  85+^  9.  5a  =  3a+14 

10.  3:c+12  =  54  11.  9x=30+4:X  12.  7o  =  45+2a. 

13.  Qa+a  =  SQ+Sa  14.  7^-^+2  =  44  15.  4:x;+:v+3  =  23 

Find  value  of: 

16.  13jc-4x-7  if  a;=2  17.  9:K4-3:r-2-6:r  if  a:=3 

18.  12a-6-2a+8  if  a  =  10         19.  13»-3+2;t-5  if  »  =  4 

27.  Subtraction  operation  repeated.   Consider  the  equation 

8+6n  =  23+w 
To  solve,  take  one  needed  step  at  a  time.     On  the  left 
side  6m  and  8  are  tangled  by  addition,  hence  to  get  rid  of  8, 
Subtract  8  from  both  sides, 

6n=15+n 
On  the  right  side  15  and  n  are  tangled  by  addition,  hence 
to  get  rid  of  n, 

Subtract  n  from  both  sides, 

5w  =  15 
n  is  tangled  with  5  by  multiplication,  hence  to  get  rid  of  5, 
Divide  both  sides  by  5,    w  =  3 


34  BEGINNERS*  ALGEBRA 

The  solution  shotild  be  put  down  as  follows: 

8+6m  =  23+w 
Subtraxjt  8  from  both  sides, 

6w  =  15+n 
Subtract  n  from  both  sides, 

5n  =  15 
Divide  both  sides  by  5,  n  =  S 
Check:* 


8+6-3 

23+3 

8+18 

23+3 

26 

26 

EXERCISES 

Solve  and  check  equations  1-10: 

1.  2a;+5  =  a;+15  2.  5w+2  =  3«+4 

3.  5w+2  =  3«+ll  4.  7^+5  =  a;+29 

5.  12o+3  =  5a+17  6.  8^+4  =  5:r+20 

7.  9iC+7  =  2ic+35  8.  7a+2  =  3a+22 

9.  10x+8  =  3jc+25  10.  14^+6  =  6a;+24 

11.  7:*:-2jc+8-3ic=?a;+?       12.  3a;+8+5:x;-2=?ic+? 

13.  2:*:+9+6ic-4  =  ?  14.  7ic+3-2«;+8  =  ? 

28.  More  complicated  equations.    Consider  the  equation 

5+6f^+3  =  28+2w 

This  equation  may  be  solved  by  three  applications  of  the 
subtraction  operation.  We  may  shorten  the  work,  however, 
by  adding  terms  within  each  side  whenever  possible  before 
using  the  subtraction  operation.  We  cannot  add  5  and  6, 
because  5+6n+3  means  5+6Xw+3  and  we  must  obey  the 
laws  of  order  (see  Art.  18).  We  may  change  the  order  of 
addition,  however,  and  add  5  and  3.     The  work  is  as  follows: 

♦We  use  the  vertical  line  to  separate  the  two  sides  to  be  checked. 


FORMULAS  AND  EQUATIONS  35 

5+6w+3  =  28+2w 
Add  5+3,  8+6w  =  28+2w 

Subtract  8  from  both  sides,  6w  =  20+2w 

Subtract  2n  from  both  sides,  4w  =  20 

Divide  both  sides  by  4,  w  =  5 

EXERCISES 

Solve  and  check: 

I.  x+Qx-\-2  =  U+Sx  2.  9w-2w+6  =  27+4« 
3.  7a+a+5  =  21+4a  4.  7o+8-6  =  26+3a 
5.  19c+9-5  =  37+8c  6.  8:r+5-3  =  5ic+17 
7.  7ic-2a;+8  =  23  8.  12+9^-3a;=48 

9.  2+ll^+3  =  7/>+25  10.  21ic-7a;+5=18+3ic 

Find  value  of: 

II.  3a-2b  if  a  =  6,  6  =  2  12.  5jcH-2y  if  x  =  i,  y  =  5 
13.  6a-26+12  if  a=5,  6  =  4      14.  9a-2+46  if  a  =  3,  6  =  5 

29.  The  addition  operation.     Five  times  a  number  less  17 
equals  8.     What  is  the  number  ?     In  algebraic  language  this  is 

5n-17  =  8 
Each  side  of  the  equation  is  17  less  than  5  times  the  num- 
ber.    If  17  be  added  to  both  sides,  the  deficiency  will  be 
made  up. 

5m  =  25 

that  is,  5^-17+17  =  8+17 

Notice  that  on  the  left  side  the  same  number  17  is  sub- 
tracted and  added ;  these  operations  destroy  each  other.    The 

solution  is  then 

5w-17  =  8 

Add  17  to  both  sides,  5w  =  25 

Divide  both  sides  by  5,         w  =  5 


Check:  5  •  5  —  17 

25-17 

8 


36 


BEGINNERS'  ALGEBRA 


We  have  thus  brought  to  light  another  operation  that  can 
be  performed  on  both  sides  of  an  equation.  Stated  in  words 
we  have 

Operation  4.  The  same  number  can  be  added  to  both 
sides  of  an  equation. 


EXERCISES 


Solve  and  check: 
1.  a;-15  =  30 
3.  a-16  =  46 
5.  7/-3  =  8 
7.  w-7-6  =  3 
9.  8;j-l  =  5w+41 

11.  6:^-4  =  24-^; 

13.  13w-6  =  8«+14 


2.  x-9  =  21 

4.  3/-5  =  15 

6.  33; -10  =  20 

8.  3^-10+2x=40 

10.  3+5a;  =  ll-2jc 

12.  Sx-21  =  2x-n 

14.  7x+15  =  2jc+25 

Although  we  usually  read  an  equation  from  left  to  right, 

it  is  sometimes  more  convenient  to  read  it  from  right  to  left 

and  keep  the  unknown  on  the  right  instead  of  on  the  left. 

For  instance, 

8-\-n=3n 

S=2n 

4:  =  n 


15.  3a+13  =  23a-7 
17.  2x+21  =  9:x-3 
19.  3a:+32-jc+2x  = 

12:x;+12-3:x: 
21.  2h  =  5h-5-2h 


16.  2:K+24=8a;4-6 

18.  4:X-]-S7-x=9x-\-9-2x 

20.  3«4-29+4w-2-2w  = 

lOw+9 
22.  4-:»  =  2 


30.  The  multiplication  operation.  The  problem:  One- 
third  of  a  number  is  4,  what  is  the  number?  can  be  answered 
very  quickly  by  arithmetic.  Simply  multiply  4  by  3. 
Translated  into  algebraic  language,  the  problem-  is 

1       . 

7rW  =  4 


n 


or 


=  4 


FORMULAS  AND  EQUATIONS  37 

for  just  as  I  of  5  may  be  written  either  as 

|x5or| 

SO  also  ^  oi  n  may  be  written 

1         n 

SO  also  f  of  w  may  be  written  either  as 

2         2n 

ft 

To  get  the  value  of  n  from  i^  =  ^y  we  multiplied  4  by  3. 

o 

To  preserve  the  balance  of  the  equation,  we  must  multiply 

the  left  side  also  by  3.     That  is, 

3. 1=3. 4 

w  =  12     (See  Art.  2) 

This  discloses  another  operation  that  can  be  used  in  hand- 
ling equations : 

Operation  5.  Both  sides  of  an  equation  may  be  multi- 
plied by  the  same  ntmiber. 

Apply  to  the  equation 

|w+8  =  15 

n  is  tangled  up  with  3  by  division,  hence  to  untangle  n, 
multiply  by  3.  That  means  both  sides  must  be  multiplied 
by  3.     Why? 

The  work  proceeds  as  follows: 

^11+8=15 

Multiply  both  sides  by  3,        n+24  =  45 
Subtract  24  from  both  sides,  n  =  2l 


38 


BEGINNERS*  ALGEBRA 


Check: 


21+8 


7-i-8 
15 


15 

15 
15 


A  more  complicated  case: 
1 


2^-2--%+^ 

To  untangle  the  x  on  left  side,  miiltiply  both  sides  by  2: 

2      2 
^-^  =  3^+3 
To  get  rid  of  the  fractions  on  the  right  side,  multiply  both 
sides  by  3 : 

3ic-12  =  2%+2 
Add  12  to  both  sides,  Zx  =  2^+14 

Subtract  2x  from  both  sides,  :x;=  14 

The  first  two  steps  may  be  taken  at  the  same  time  by 
multiplying  both  sides  by  2X3  =  6. 


Thus 


2^-2  =  3^+3 


Multiply  both  sides  by  6,     3:\;  - 12  =  2;t;+2 
Add  12  to  both  sides,  Sx  =  2ic+ 14 

Subtract  2x  from  both  sides,        x  =  14i 


Check: 

1.U-. 

1- 

7-2 

14,1 
3^3 

5 

15 
3 

5 

5 

EXERCISE  I 

Find  value  of: 

1.  4\x+3\  if  x=2,  x=S,  x  = 


10 


FORMULAS  AND  EQUATIONS 

2,  3(^+5)  ^  x=2,  x=5,  x=7 

3.  e^l+s)  if  a=2,  fl=5,  a=7 


4.  Show  that  6(  ^+5  |  =  2a+30.     Find  value   of  both   sides  if 
a=2;  if  a=7. 

5.  Show  that  10(  — +7  )  =  4:\:+70.     Find  value  it  x=3:ii  x=5. 


6.  Are  the  statements  in  Exercises  4  and  5  true  for  any  values  of 
a  and  x  that  yo*u  may  choose? 


EXERCISE 

II 

Solve  and  check 

:: 

1.  |.=5 

2.  |„=12 

3.  ^=3 

4.  g=15 

5.f=16 

6.  ^n+n  =  9 

7.  f+»=12 

8.  /4=6 

«•  ri+« 

10.  ia-2=| 

11.  w-^  =  J 

S 

-l+l- 

13.  ^+1=5 

14    '-+'-'-= 
^*-  4^2     5 

=  11 

1^- 1=1 

16.  1^-5=1)- 

1 
■3 

17    4f_2_^ 

2 
-5^ 

18.  7w-5»+i»  = 

5 

6 

19.  One-third  of  a  number  increased  by  one-half  the  same  num- 
ber equals  35.    What  is  the  number? 

31.  Summary  of  allowable  operations  on  equations.    In 

the  preceding  articles  we  have  been  learning  how  to  solve 
equations.  Our  effort  has  been  to  untangle  the  unknown 
number  from  all  the  known  numbers  and  thus  determine 
its  actual  value.  In  doing  this  we  have  found  five  operations 
that  can  be  used  upon  an  equation  without  disturbing  the 
balance  of  the  equation.     These  operations  do  not  in  any 


40  BEGINNERS'  ALGEBRA 

way  change  the  value  of  the  unknown  number;  they  merely 
get  rid  of  certain  known  numbers  that  are  in  positions  where 
they  are  not  desired.  We  may  use  any  of  the  five  operations 
that  seem  necessary  with  the  certainty  that  the  number 
sought  will  be  found  if  there  is  such  a  number. 

Operation  1.  Either  side  of  an  equation  may  be  altered 
in  form  by  the  performance  of  any  of  the  indicated  operations 
that  are  possible. 

Operation  2.  Both  sides  may  be  divided  by  the  same 
known  number,  excepting  the  number  zero.* 

Operation  3.  The  same  number  may  be  subtracted  from 
both  sides. 

Operation  4.  The  same  ntimber  may  be  added  to  both 
sides. 

Operation  5.  Both  sides  may  be  multiplied  by  the  same 
known  number. 

We  shall  find  later  that  there  are  other  operations  that  can 
be  used  in  solving  equations,  but  we  do  not  need  to  consider 
them  here. 

The  operations  to  be  used  in  solving  any  given  equation 
are  the  ones  that  will  do  what  is  needed  to  be  done.  There 
is  no  particular  order  in  which  these  operations  should  be 
used.  Simply  decide  what  needs  to  be  done  and  then  choose 
the  operation  that  accomplishes  your  purpose.  Often  a 
thoughtful  choice  of  one  operation  rather  than  another  will 
greatly  reduce  the  amount  of  work  necessary. 

Illustration.  bx—7—Zx=2X 

Here  it  is  best  first  to  make  any  alteration  in  the  left  side  that 
may  be  possible;  that  is,  Zx  can  be  subtracted  from  bx,  and  we  have 

2:^-7  =  21 
We  theii  need  to  get  rid  of  the  term  7,  and  we  can  do  this  by 
adding  7  to  both  sides: 

;^x=28 

♦Division  by  zero  is  not  permitted  in  mathematics. 


FORMULAS  AND  EQUATIONS 


41 


To  get  rid  of  the  coefficient  2,  use  division: 

^  =  14 
Check: 


5x14-7-3x14 

21 

70-7-42 

21 

63-42 

21 

21 

21 

MISCELLANEOUS   EXERCISES 
Solve  and  check.     In  each  case  indicate  the  operation  used. 


1.  6w+5  =  47 

2. 

7+;j  =  31 

3.  3«-i-2w+16  =  31 

4. 

5»+3»+10  =  90 

«•  1+1=^ 

6. 

46-3w  =  7 

7.  o-\-pi  =  n 

8. 

32=Jx 

9.48-4. 

10. 

w+5  =  ll 

11.  154-&=40 

12. 

9»-3»  =  66 

13.  4w+w  =  4 

14. 

8.-2  =  6:c+6 

15.  9^-5  =  6w-2 

16. 

2w-5-w-3  =  0 

17.  18-w=«-18 

18. 

5+4:r  =  2a:4-10 

19.  4«+5  =  5«+5 

20. 

5-33'=>'-l 

21.  6x-4  =  3:r-2 

22. 

15-7/=9/+3 

23.  2.-8+3:r=a:+12 

24. 

|.+5  =  |.+11 

25.  \x-2  =  ^-^l 

26. 

7A+5+2/?-2  =  ^ 

4A+3 

Suggestion.     It  is  often  better  to  add  a  term  to  both  sides  or  to 
subtract  a  term  from  both  sides  before  any  attempt  is  made  to  com- 
bine terms  on  the  same  side.    In  some  cases  this  must  be  done. 
7+3a:-3  =  ac-34-15 
Add  3  to  both  sides,       7 + 3a; = ac + 15 
Then  Zx  =  x+^ 

2a;  =  8 
..  =  4 
27.  15ic+42  =  3+10ic+42  28.  llx+13-3:x:=29-3a: 

29.  15a -42 -7a= 93 -7a -50        30.  3+2:«-6=l 
31.  4a+7-5a  =  6a-42 

4 


CHAPTER  III 
The  Solution  of  Problems 

32.  Introductory.  One  of  the  most  important  uses  of 
algebra  is  the  solution  of  problems.  In  fact,  the  algebraic 
language  has  been  developed  in  an  effort  to  simplify  the 
working  out  of  problems.  When  a  problem  is  presented,  the 
first  thing  to  be  done  is  to  translate  its  ideas  into  algebraic 
shorthand. 

33.  Translations.  In  translating  from  English  words  into 
algebraic  symbols  notice: 

(1)  An  Arabic  figiure  is  used  to  stand  for  a  known  number. 

(2)  A  letter  is  used  to  stand  for  an  imknown  number. 

(3)  The  plus  sign,  +,  is  used  to  express  a  sum. 

(4)  The  minus  sign,  — ,  is  used  ta  express  a  difference. 

(5)  In  the  writing  of  products  the  factors  are  written  close 
together  without  a  sign  between  them:  la  instead  of  7Xa. 
If,  however,  we  have  the  product  of  two  known  numbers, 
a  sign  must  be  used  between  them:  7X6  or  7  •  6. 

(6)  The  quotient  of  two  numbers  is  written  as  a  fraction, 
4  a 

5' 5* 

34.  Exercises  in  translation.  State  the  following  ideas  in 
algebraic  language: 

1.  Three  added  to  an  unknown  number. 

2.  An  unknown  number  minus  5. 

3.  Eleven  times  a  number. 

4.  One-fifth  of  a  number. 

5.  Nine  divided  by  an  unknown  number. 

6.  The  quotient  of  7  and  an  unknown  number. 

7.  One-sixth  of  5  times  a  number. 

42 


THE  SOLUTION   OF   PROBLEMS  43 

8.  The  sum  of  an  unknown  number  and  7. 

9.  The  sum  of  a  nimiber  and  7,  divided  by  9. 
10.  A  number  plus  7,  divided  by  9. 

IL  The  difference  between  a  number  and  5. 

12.  The  difference  between  a  nimiber  and  7,  divided  by  7. 

13.  Three  times  a  number,  plus  7. 

14.  Three  times  the  sum  of  a  number  and  8. 

15.  The  ratio  between  two  numbers. 

16.  Three  times  the  difference  between  a  number  and  4. 

When  it  is  necessary  to  multiply  or  to  divide  a  sum  or  a 
difference  by  any  number,  inclose  the  sum  or  the  difference 
in  parentheses;  for  example,  4(^+1)  represents  4  times  the 
simi  of  a  number  and  one. 

17.  Translate  the  following  into  English,  using  the  words  "stmi," 
"difference,"  "product,"  and  "quotient"  wherever  they  are  called 
for: 


(a)  4+a 

(b)  5x 

«>i 

(^1 

(e)  a-1 

(/)  66+2 

(g)  S^2x 

ih)  a+l 

^'^      5 

U)  5(a+4) 

(k)  5a+4 

(/)  5a+2b 

(m)  5fl-4 

(n)  5(a-4) 

(P)    lb 

(q)  (1-^)5 

W  x-2 

18.  Evaluate  each  of  the  expressions  given  in  Exercise  17,  using 
a=10,  6  =  6,  x  =  2. 

Translate  the  following  into  algebraic  language: 

19.  A  man  is  y  years  old.    How  old  will  he  be  in  3  years? 

20.  A  boy  is  x  years  old.    How  old  was  he  5  years  ago? 

21.  If  a  boy  receives  x  cents  a  day,  how  much  will  he  receive  in  a 
month? 


44  BEGINNERS'  ALGEBRA 

22.  How  many  feet  in  6  yards?    How  many  feet  in  b  yards? 

23.  A  boy  earns  x  dollars  a  week.  How  much  does  he  earn  in  10 
weeks?  How  much  does  he  receive  in  10  weeks  if  he  receives  a 
raise  of  one  dollar  a  week? 

24.  What  is  the  average  of  3,  10,  17? 

25.  What  is  the  average  of  a,  b,  c? 

26.  If  sweaters  sell  at  wholesale  at  $75  a  dozen,  what  is  the  cost 
price  of  one  sweater?    Of  b  sweaters? 

27.  If  the  wholesale  price  of  the  sweaters  in  Exercise  26  is 
increased  x  dollars  a  dozen,  what  will  be  the  cost  of  one  sweater 
at  the  increased  price? 

28.  What  is  the  reciprocal  of  2?    Of  7?    Of  I ?    Of  ri ? 

29.  What  is  the  reciprocal  of  a? 

30.  If  the  difference  between  two  numbers  is  5  and  the  larger 
is  represented  by  a,  what  will  represent  the  other?  What  will 
represent  their  sum?    Their  product?    Their  quotient? 

31.  The  weight  of  a  cup  containing  250  shot  is  W;  the  weight  of 
the  empty  cup  is  w.    What  will  express  the  weight  of  one  shot? 

32.  The  dimensions  of  one  wall  of  a  room  are  x  and  y  feet.  What 
will  express  the  area  of  the  wall?  If  the  wall  has  3  windows  each 
ahy  b  feet,  what  will  express  the  area  of  each  window?  Of  all  3 
windows?     Of  the  wall  space? 

Translate  into  words  the  following  algebraic  statements: 

35.  2x-3 


33.  Sa 

34.  5o+7 

36.     3 

37.  5-n 

39.  3(:^-2) 

40.5 

n 

42.  10w-h3(w- 

-1) 

43.  100-^ 

o 

38. 

2/+5/- 

-3 

41. 

n 

44. 

n-\-2 
n-2 

35.  Problems  relating  to  numbers.  In  trying  to  solve  the 
following  problems:  (a)  translate  the  sentence  into  an 
equation,  using  some  letter  for  the  unknown;  (6)  solve  the 
equation;  (c)  check  the  result. 


THE   SOLUTION   OF  PROBLEMS  45 

Problems: 

1.  The  sum  of  an  unknown  number  and  8  is  23.     Find  the 
nimiber. 

2.  The  difference  between  an  unknown  number  and  10  is  2.     Find 
the  nimiber.     Can  there  be  more  than  one  answer. 

3.  The  sum  of  3  times  a  number  and  2  is  17.     Find  the  number. 

4.  Ten  times  a  ntunber  minus  2  is  23.     Find  the  number. 

5.  The  sum  of  three-fourths  of  a  number  and  7  is  16.     Find  the 
number. 

6.  Three-fourths  of  a  number  minus  6  is  9.     Find  the  number. 

7.  If  5  is  added  to  6  times  a  number,  the  result  is  47.    What  is 
the  ntunber? 

8.  The  sum  of  7  and  a  certain  number  is  31.     What  is  the 
unknown  ntunber? 

9.  If  3  times  a  number,  twice  the  number,  and  16  are  added,  the 
result  is  31.    Find  the  number. 

10.  The  sum  of  5  times  a  number  and  3  times  the  number  and 
10  is  90.    What  is  the  number? 

11.  The  sum  of  a  third  of  a  number  and  a  fifth  of  a  number  is  24. 
What  is  the  number? 

12.  There  is  a  number  such  that  3  times  the  number  subtracted 
from  46  is  7.    What  is  the  number? 

13.  If  a  foiuth  of  a  certain  number  is  added  to  5,  the  result  will 
be  the  number  itself.    What  is  the  number? 

14.  Thirty-two  is  one-fifth  of  what  number? 

15.  Forty-eight  is  5  per  cent  of  what  number? 

16.  A  number  increased  by  5  equals  11.     Find  the  number. 

17.  A  ball  and  a  bat  together  cost  40  cents;  the  bat  cost  15  cents. 
What  did  the  ball  cost? 

18.  The  difference  between  3  times  a  certain  number  and  9  times 
the  same  number  is  66.    What  is  the  nimiber? 

19.  The  sum  of  4  times  a  number  and  the  number  is  4.    What  is 
the  number? 


46  BEGINNERS'  ALGEBRA 

Problems  are  not  always  so  easily  stated  as  the  preceding, 
which  are  almost  word-for-word  translations. 

The  sum  of  two  numbers  is  78;  one  of  the  numbers  is  5 
times  the  other.     What  are  the  numbers? 

The  equality  that  must  be  stated  is  obvious : 

one  number + other  number  =  78 

Both  numbers  are  unknown;  we  will  choose  one  of  them 
as  the  unknown  of  the  equation  and  represent  it  by  w.  The 
problem  states  that  one  is  5  times  the  other,  consequently 
the  second  number  is  5w,  and  the  equation  becomes 

w-f5»  =  78 

6w  =  78  I 

n  =  13,  one  number 
5w  =  65,  the  other  number 

20.  The  sum  of  two  ntimbers  is  72.  One  of  them  is  6  less  than 
the  other.    What  are  the  numbers? 

21.  The  larger  of  two  nimibers  is  27.  Their  difference  is  13. 
What  are  the  numbers? 

22.  Find  the  value  of  the  following  if  ic=5: 

(a)  x+U+x  (b)  Sx+S-x 

(c)  4cX-12-x  (d)  6:«-3+2:x; 

(e)  x+4:+2x+8  (/)  x+x+Sx+5 

23.  State  each  of  the  expressions  in  Exercise  22  in  a  shorter  form. 

24.  In  a  class  of  48  pupils  there  are  twice  as  many  girls  as  boys. 
How  many  are  there  of  each? 

25.  Separate  100  into  two  parts  one  of  which  shall  be  4  times  the 
other. 

26.  A  horse  and  a  wagon  cost  $540.    What  was  the  cost  of  each 

if  the  wagon  cost  twice  as  much  as  the  horse? 

27.  Tom,  Sam,  and  John  have  63  cents.  Sam  has  twice  as  many 
as  Tom,  and  John  has  twice  as  many  as  Sam.  How  many  does 
each  have? 


THE  SOLUTION   OF  PROBLEMS  47 

28.  Two  hundred  and  seventy  dollars  is  divided  among  3  chil- 
dren. The  second  receives  twice  as  much  as  the  first,  while  the 
third  receives  twice  as  much  as  the  other  two.  What  was  the  share 
of  each? 

36.  Solving  problems.  In  each  problem  to  be  solved 
certain  steps  are  to  be  taken: 

(1)  Find  an  equality. 

(2)  Choose  an  unknown  of  the  problem  to  be  the  unknown 
of  the  equation  and  represent  it  by  a  letter. 

(3)  Restate,  or  translate,  the  equality  in  algebraic  sym- 
bols, using  the  letter  for  the  unknown.  In  the  problems  of 
this  chapter  it  will  be  found  that  after  one  of  the  unknowns 
of  the  problem  is  expressed  by  a  letter  the  other  unknown 
can  be  expressed  by  that  letter  and  the  other  known  number 
of  the  problem. 

(4)  Solve  the  equation. 

(5)  Check,  so  as  to  be  sure  that  the  working  out  of  the 
solution  has  been  done  without  mistake. 

(6)  Verify  by  applying  the  numbers  to  the  problem  itself 
to  see  if  they  actually  work. 

Illustration.  Three  times  a  certain  even  ntunber  plus  7  is  22. 
Find  the  number. 

(1)  Simply  translate  into  algebraic  language. 

(2)  Let  X  represent  the  imknown. 


(3)  The  equation  is 

Sx-\-7  = 

=  22 

(4)  The  solution  is 

3^=15 

x=5 

(6)  Check: 

3x5+7 

15+7 

22 
22 

22    22 

The  equation  has  been  worked  out  without  mistake. 
(6)  Verification.    The  answer  5  will  not  work  in  the  problem, 
for  it  is  not  an  even  number. 


48  BEGINNERS'  ALGEBRA 

37.  Rectangle  problems.     Put  the  following  into  alge- 
braic form: 

Translations: 

1.  If  one  side  of  a  rectangle  is  5  and  the  other  is  unknown,  what 
will  represent  the  perimeter?    What  will  represent  the  area?  (Fig.  4.) 

2.  If  one  side  of  a  rectangle  is  3  inches  longer 
than  the  other,  what  will  represent  the  per- 
imeter?   What  will  represent  the  area? 


Pjq   4  3.  If  one  side  of  a  rectangle  is  2  inches  shorter 

than  the  other,  what  will  represent  the  per- 
imeter?   What  will  represent  the  area? 

4.  If  two  sides  of  a  rectangle  are  in  the  ratio  2  to  1,  write  a 
number  to  express  each  side. 

5.  If  two  sides  of  a  rectangle  are  in  the  ratio  2  to  3,  write  a  number 
to  express  each  side. 

Problems: 

1.  It  takes  410  feet  of  wire  fencing  to  inclose  a  lot  that  is  172  feet 
deep.     How  wide  is  the  lot? 

The  equality  is,  the  perimeter  =  410 

One  side  is  known,  use  x  for  the  unknown  side 

The  equation  is  then  ic+172-|-:v+172  =  410 

Let  the  pupil  solve  the  equation.  Check  and  verifj^  Can  the 
perimeter  be  expressed  in  a  more  condensed  form? 

2.  It  takes  420  feet  of  fence  to  inclose  a  square  lot.  Find  one 
side  of  the  lot. 

3.  The  length  of  a  rectangular  field  is  3  times  the  width.  It 
requires  840  feet  of  fence  to  inclose  it.  What  are  the  dimensions 
of  the  field? 

4.  If  the  perimeter  of  a  rectangle  is  122  feet  and  the  width  is 
21  feet,  find  the  length.  What  will  represent  the  perimeter?  The 
area? 

5.  The  length  of  a  rectangle  is  3  feet  more  than  its  width.  If 
the  perimeter  is  42  feet,  find  the  dimensions. 


THE  SOLUTION   OF   PROBLEMS  49 

6.  The  length  of  a  rectangle  is  2^  times  the  width.    What  are 
the  dimensions  if  the  perimeter  is  49? 

7.  The  width  of  a  certain  rectangle  is  f  of  its  length.     If  the 
perimeter  is  110  inches,  find  the  dimensions. 

8.  The  perimeter  of  a  rectangle  is  130  feet.     If  the  length  is 
5  feet  more  than  3  times  the  width,  find  the  dimensions. 

9.  The  perimeter  of  a  rectangle  is  130  feet.     If  the  length  is 
5  feet  less  than  its  width,  find  the  dimensions. 

10.  The  perimeter  of  a  certain  rectangle  is  70  feet.     If  the  length 
is  5  feet  less  than  3  times  the  width,  find  the  dimensions. 

11.  Two  sides  of  a  rectangle  are  in  the  ratio  2  to  1 ;  the  perimeter 
is  36.    What  are  the  dimensions  of  the  rectangle? 

12.  Two  sides  of  a  rectangle  are  in  the  ratio  2  to  3.    If  the 
perimeter  is  00,  what  are  the  dimensions  of  the  rectangle? 

13.  Express  the  following  in  shorter  form: 

(a)  :c+3x+5+2a;+10  (b)  3;c-ic+5+2«-2 

(c)  6a;+10~3C-5+2;c  (d)  3jc+2+4jc4-5 

14.  Find  the  value  of  the  expressions  in  Exercise  13  if  a; =3. 

38.  Triangle  problems.  The  triangle  receives  its  name 
from  the  fact  that  it  has  3  comers  or  angles.  A  rectangle  has 
4  angles  and  receives  its  name  from  the  fact  that  its  angles 
are  all  right  angles.  That  the  sum  of  the  angles  of  a 
rectangle  is  4  right  angles  is  easily  seen  from  Fig.  5.  It  is 
a  rather  remarkable  fact 
that  the  sum  of  the  angles 
of  a  triangle  is  2  right 
angles.  As  a  right  angle 
is  too  large  an  angle  to  be  Fig.  5 

a  convenient  unit  for 

measuring  angles,  it  is  divided  into  90  equal  parts,  each  of 
which  is  called  a  degree.  Hence  a  right  angle  contains  90 
degrees.  We  may  say,  then,  that  the  sum  of  the  angles  of 
a  triangle  is  180  degrees.     If  we  use  a,  b,  and  c  for  the 


50  BEGINNERS'  ALGEBRA 

number  of  degrees  in  each  of  the  angles  of  a  triangle,  we 
may  state  this  very  important  fact  in  the  algebraic  form 

a+6+c=180 

You  may  verify  this  equality  by  tearing  off  the  corners 
and  rearranging. 

Translations: 

1.  If  the  sum  of  two  angles  of  a  triangle  is  s,  what  will  represent 
the  other  angle? 

2.  If  one  angle  of  a  triangle  is  w,  what  will  represent  the  sum  of 
the  other  two? 

3.  If  a  triangle  has  three  equal  sides,  what  will  represent  its 
perimeter? 

Problems: 

1.  The  perimeter  of  a  triangle  is  105  inches.  What  is  the  third 
side  if  two  of  the  sides  are  30  inches  and  40  inches? 

2.  One  side  of  a  triangle  is  2  feet  more  than  a  second,  the  third 
is  3  feet  more  than  the  second,  and  the  perimeter  is  50  feet.  Find 
the  length  of  each  side. 

3.  One  side  of  a  triangle  is  twice  a  second,  and  the  third  is 
1^  times  the  second.  If  the  perimeter  is  18  feet,  find  the  length  of 
each  side. 

4.  Two  sides  of  a  triangle  are  equal,  the  third  side  is  4,  and  the 
perimeter  is  16.    Find  the  sides  of  the  triangle. 

5.  Two  sides  of  a  triangle  are  equal,  the  third  side  is  ^  of  one 
of  the  equal  sides,  and  the  perimeter  is  30.  Find  the  sides  of  the 
triangle. 

6.  The  sides  of  a  certain  triangle  are  all  equal.  If  the  sides  in 
succession  are  increased  by  1,  2,  3,  the  perimeter  of  the  resulting 
triangle  will  be  12.     Find  the  sides  of  the  triangles. 

7.  One  angle  of  a  triangle  is  twice  a  second,  and  the  third  is  3 
times  the  second.     Find  the  angles  of  the  triangle. 

8.  One  angle  of  a  triangle  is  20  degrees  more  than  a  second,  and 
the  third  is  twice  the  second.    Find  the  angles. 


THE  SOLUTION   OF  PROBLEMS  51 

9.  If  one  angle  of  a  triangle  is  42  degrees  more  than  a  second  and 
10  degrees  more  than  the  third,  find  the  number  of  degrees  in  each 
angle. 

10.  If  one  angle  of  a  triangle  is  20  degrees  less  than  a  second 
and  20  degrees  more  than  the  third,  find  the  number  of  degrees  in 
each  angle. 

11.  One  angle  of  a  triangle  is  25  degrees  more  than  a  second, 
and  the  third  is  15  degrees  more  than  twice  the  second.  Find  the 
number  of  degrees  in  each  angle. 

12.  One  angle  of  a  triangle  is  f  of  a  second,  and  the  third  is 
4  degrees  less  than  the  second.  Find  the  number  of  degrees  in 
each  angle. 

13.  One  angle  of  a  triangle  is  30  degrees,  and  the  other  two  angles 
are  equal.     Find  the  niunber  of  degrees  in  each  angle. 

14.  One  of  the  two  equal  angles  of  a  certain  triangle  is  ^  of  the 
third  angle.     Find  all  the  angles. 

39.  Digit  problems.  In  a  number  of  two  figures  or  digits, 
what  does  the  right-hand  figure  mean?  What  does  the 
left-hand  figure  mean  ?  If  the  left-hand  digit  is  2  and  the 
right-hand  digit  is  6,  how  is  the  value  of  the  number  obtained  ? 

Translations: 

1.  The  left-hand  digit  is  2  and  the  right-hand  digit  is  a.  Repre- 
sent the  number. 

2.  The  left-hand  figure  is  a  and  the  right-hand  figure  is  5. 
Represent  the  nimiber. 

3.  The  left-hand  figure  is  a  and  the  right-hand  figure  is  b.  Repre- 
sent the  number. 

4.  Write  a  number  of  three  digits  in  which  the  digits  beginning 
at  the  left  are  2,  3,  and  a. 

5.  Write  a  mmiber  of  three  digits  in  which  the  digits  beginning 
at  the  right  are  (1)  5,  7,  a;  (2)  b,  7,  a;  (3)  7,  c,  5. 

6.  Write  a  number  whose  digits  from  left  to  right  are  ac,  y,  z. 


52  BEGINNERS'   ALGEBRA 

7.  Write  a  number  of  three  digits  in  which  the  digits  from  left 
to  right  are  a,  0,  and  3. 

8.  If  the  units  digit  of  a  ntunber  is  5  less  than  the  tens  digit,  how 
may  the  number  be  represented? 

9.  If  the  tens  digit  of  a  number  is  3  more  than  the  units  digit, 
how  will  you  represent  the  number? 

Problems: 

1.  In  a  nimiber  of  2  digits,  the  tens  digit  is  3  more  than  the  units 
digit.     If  the  stun  of  the  digits  is  15,  find  the  number. 

2.  In  a  number  of  2  digits,  the  units  digit  is  4  more  than  the  tens 
digit.     If  the  sum  of  the  digits  is  14,  what  is  the  number? 

3.  The  units  digit  of  a  certain  number  is  f  of  the  tens  digit.  The 
difference  between  the  two  digits  is  2.    What  is  the  number? 

4.  The  right-hand  digit  of  a  certain  number  is  4  less  than  the 
left-hand  digit.  The  number  is  6  more  than  9  times  the  left-hand 
digit.     Find  the  number. 

5.  The  right-hand  digit  of  a  certain  number  is  one  more  than  the 
left-hand  digit.  The  number  is  equal  to  12  times  the  left-hand 
digit  minus  5.    Find  the  number. 

6.  The  units  digit  of  a  number  is  twice  the  tens  digit.  The  num- 
ber is  equal  to  4  more  than  11  times  the  tens  digit.  Find  the 
number. 

7.  The  units  digit  of  a  certain  number  is  3  times  the  tens  digit. 
The  niunber  is  equal  to  5  times  the  units  digit  minus  6.  Find  the 
number. 

8.  The  units  digit  of  a  certain  munber  is  3  times  the  tens  digit. 
The  number  is  5  more  than  11  times  the  tens  digit.  What  is  the 
number? 

40.  Consecutive  integer  problems.  Give  illustrations  of 
two  consecutive  integers;  of  two  consecutive  fractions. 

Translations: 

1.  If  «  is  an  integer,  what  is  the  next  higher  integer?  The  next 
lower  integer? 


THE  SOLUTION  OF  PROBLEMS  53 

2.  K  «  is  an  integer,  write  an  expression  for  an  even  number; 
for  an  odd  number. 

3.  Write  expressions  for  three  consecutive  integers. 
Problems: 

1.  The  sum  of  two  consecutive  integers  is  99.    What  are  the 

integers? 

2.  Find  three  consecutive  integers  whose  sum  is  63. 

3.  Find  four  consecutive  integers  whose  sum  is  126. 

41.  Coin  problems.  In  solving  coin  problems  keep  in 
mind  the  relation  of  the  values  of  the  coins  to  one  another. 

Translations: 

1.  How  many  cents  in  d  dimes?    In  q  quarters?    In  n  nickels? 

2.  How  many  nickels  in  d  dimes?    In  q  quarters?    In  c  cents? 

3.  How  many  quarters  in  c  cents?     In  n  nickels? 

4.  Write  an  expression  to  represent  the  number  of  cents  in  n 
nickels  and  2  cents;  in  d  dimes  and  7  cents;  in  d  dimes  and  n  nickels 
and  c  cents. 

Problems: 

1.  I  have  72  cents  in  nickels  and  one-cent  pieces,  and  have  the 
same  number  of  each.    How  many  nickels  have  I? 

Suggestion.  The  value  of  one-cent  pieces  plus  the  value  of  the 
nickels  equals  72  cents. 

2.  I  have  the  same  number  of  nickels  and  dimes,  amounting  in 
aU  to  $1 .  05.    What  is  the  number  of  dimes? 

3.  In  a  pocket  full  of  change  amoimting  in  all  to  $4.05  there  are 
twice  as  many  dimes  as  quarters.     How  many  are  there  of  each? 

4.  I  bought  some  3-cent  stamps  and  twice  as  many  2-cent  stamps. 
If  I  paid  for  them  all  70  cents,  how  many  of  each  did  I  buy? 

5.  I  have  SI  cents  in  quarters  and  one-cent  pieces.  If  I  have  3 
more  one-cent  pieces  than  I  have  quarters,  how  many  have  I  of  each? 

6.  I  have  98  cents  in  nickels  and  one-cent  pieces.  If  I  have  2 
more  one-cent  pieces  than  I  have  nickels,  how  many  have  I  of  each? 


64  BEGINNERS^  ALGEBRA 

7.  I  have  95  cents  in  nickels  and  dimes.  If  I  have  one  more 
nickel  than  I  have  dimes,  how  many  have  I  of  each? 

42.  Speed  problems.  If  a  cyclist  rides  18  miles  in  2 
hours,  we  say  that  his  average  rate  of  speed  is  -^  miles  per 
hour.     In  other  words, 

distance  - 

—-. =speed  d^st 

tmie 

Translations: 

1.  A  man  walks  at  the  rate  of  4  miles  an  hour.  How  far  can  he 
walk  in  n  hovirs? 

2.  How  long  will  it  take  the  man  mentioned  in  Exercise  1  to  walk 
m  miles? 

3.  How  long  will  it  take  a  man  to  cycle  x  miles  at  the  rate  of 
15  miles  an  hour? 

4.  If  10  gallons  of  water  flow  from  a  pipe  in  one  hour,  how  much 
will  flow  from  it  in  7  hours?    In  b  hours?    In  one  day?    In  x  days? 

5.  If  a  train  moves  at  the  rate  of  r  miles  an  hour,  how  far  will  it 
move  in  3  hours?    In  /  hours? 

6.  If  a  train  moves  at  the  average  rate  of  r  miles  an  hour,  how 
long  will  it  take  it  to  travel  30  miles?    To  travel  m  miles? 

7.  What  is  the  average  rate  of  a  train  that  travels  36  miles  in 
2  hours?  That  travels  m  miles  in  3  hours?  That  travels  60  miles 
in  /  hoiirs?    That  travels  m  miles  in  /  hours? 

8.  A  man  can  row  in  still  water  at  the  rate  of  6  miles  an  hour. 
A  river  is  flowing  at  the  rate  of  x  miles  an  hour.  How  far  can  he 
row  downstream  in  one  hour?  How  far  can  he  row  upstream  ixi 
one  hour? 

9.  If  a  man  can  row  at  the  rate  of  r  miles  an  hour  in  still  water, 
what  will  be  his  rate  rowing  downstream  if  the  current  is  2  miles 
an  hour?    What  is  his  rate  upstream? 

Problems: 

1.  A  man  makes  a  journey  of  147  miles.  He  cycles  5  times  as 
many  miles  as  he  walks,  and  rides  on  a  train  15  miles  more  than 
he  cycles.    How  far  does  he  ride  on  the  train? 


THE  SOLUTION  OP  PROBLEMS  55 

2.  A  certain  man  can  cycle  5  times  as  fast  as  he  can  walk.  How 
many  miles  does  he  walk  in  one  hour  if  he  makes  a  journey  of 
69  miles,  cycling  4  hours  and  walking  3  hours? 

3.  A  man  started  for  a  place  28  miles  away.  He  traveled  IJ 
hours  by  cycle  when  an  accident  occurred,  and  he  walked  for  the 
rest  of  the  way.  He  finished  the  trip  in  4  hours.  At  what  rate 
did  he  walk  if  he  cycled  at  three  times  his  walking  rate? 

4.  Two  pipes  supply  a  cistern  which  can  hold  150  gallons  of 
water.  One  supplies  2  gallons  per  minute,  the  other  3  gallons  per 
minute.     How  soon  will  the  cistern  be  full? 

5.  Two  pipes  supply  a  cistern  which  can  hold  180  gallons  of 
water.  One  pipe  supplies  2  gallons  per  minute,  the  other  5  gallons 
per  minute.  At  the  same  time  water  is  being  pimiped  from  the 
cistern  at  the  rate  of  4  gallons  per  minute.  How  long  will  it  take 
to  fill  the  cistern? 

6.  A  certain  tank  holds  30  gallons  of  water.  Two  pipes  flow 
into  it.  One  pipe  supplies  3  gallons  more  per  minute  than  the 
other.  If  the  tank  is  filled  when  the  larger  pipe  is  open  2  minutes 
and  the  smaller  pipe  4  minutes,  how  much  water  flows  through 
each  pipe  per  minute? 

43.  Generalized  statements.  Number  S3rmbols,  special, 
general.  When  we  wish  to  represent  on  paper  the  niimber 
five,  we  usually  write  the  figure  5,  though  sometimes  for 
special  purposes  we  use  the  fomis  V,  :  • :  It  is  interesting 
to  notice  that  men  have  not  always  written  numbers  as  we 
do  now.  A  very  ancient  and  a  very  natural  way  of  writing 
five  was  |  1 1 1 1 .     Other  forms  are  given  in  Fig.  6. 


','.'  "1 

IX 

€ 

V 

9 

Early  Egyptian 

Ancient  Hindu 

Fig.  6 

Greek 

Roman 

West  Arabian 

Most  of  these  systems  of  writing  numbers  were  so  incon- 
venient   for    making    arithmetical    calculations    that    the 


56  BEGINNERS'  ALGEBRA 

ancients  for  the  most  part  did  their  figuring  on  their  fingers, 
on  the  abacus,  or  on  the  dustboard.  The  multiplication  of 
XIX  by  XIII  is  very  difficidt  when  the  Roman  notation  is 
used,  but  very  simple  with  our  present  notation  19  by  13. 
Some  time  during  the  Middle  Ages  the  Arabs  introduced 
into  Europe  the  notation  we  now  use:  the  ten  symbols, 
0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  with  the  very  important  idea  of 
place  value,  by  which  we  write  23  for  twenty-three.  They 
learned  it  from  the  Hindus.  The  story  of  the  origin  and 
development  of  this  way  of  writing  numbers  is  of  great 
interest.  It  is  known  that  the  Hindus  used  nine  of  these 
figures,  all  except  0,  as  early  as  631  a.d.  It  may  be  that  the 
origin  of  some  of  the  symbols  can  be  traced  back  to  the 
Babylonians,  who  lived  thousands  of  years  before  that  time. 

As  long  as  we  wish  to  write  down  only  definite  fixed 
numbers,  the  Hindu  notation  serves  our  purpose  admirably. 
But  if  we  wish  to  write  down  a  number  whose  particular 
value  is  unknown,  or  to  write  down  the  idea  "any  number" 
without  having  in  mind  any  particular  number,  the  Hindu 
notation  fails  us.  We  have  seen  how  we  may  use  a  letter 
of  the  alphabet  to  represent  some  unknown  nimiber  whose 
value  we  wish  to  find.  In  some  of  the  formulas  of  chapter  ii 
a  letter  was  used  to  represent  *  *  any  ntmiber . ' '  For  instance, 
in  the  formula  for  the  area  of  a  rectangle 

a  =  lw 
I  stands  for  any  niimber.     So  also  w  stands  for  any  number. 
Two  different  letters  are  used  to  indicate  that  the  two  dimen- 
sions need  not  be  the  same. 

A  letter,  then,  such  as  a,  can  be  used  to  stand  for  any 
number  whatsoever. 

a +6  means  the  sura  of  any  two  numbers. 

T  means  the  quotient  of  any  two  numbers„ 

What  does  ac  mean? 
What  does  a—c  mean? 


EXERCISES 

State  in  words  the 

meaning  of: 

1    a+b 

2.  o+J+c 

^    a-\-h    a-b 
*•     2     '     2 

5.  x—y 

7.  a(x-y) 

8.  ax+bx 

10.  f 

11.  {x-y)b 

THE  SOLUTION   OF  PROBLEMS  57 


o   a+b 
6.  «3'+- 

y 

9.  a6c 

12.  afec^j; 

Translate  into  algebraic  symbols: 

13.  The  sum  of  three  ntimbers. 

14.  Twice  the  difference  between  two  numbers. 

15.  The  product  of  a  number  and  itself. 

16.  The  product  of  two  numbers  divided  by  a  third  nimiber. 

17.  The  product  of  a  number  and  the  sum  of  two  other  numbers. 

18.  A  ntmiber  divided  by  the  difference  between  two  other 
numbers. 

19.  Any  number  that  is  divisible  by  2. 

20.  The  area  of  any  rectangle. 

21.  The  volimie  of  a  brick. 

22.  The  area  of  any  triangle. 

23.  The  average  of  any  three  numbers. 

24.  The  surface  of  a  brick. 

25.  The  surface  of  a  cube. 

26.  The  voltmie  of  a  cube. 

27-38.    Find  the  value  of  Exercises  1-12  when  a=17,  &=9, 
:r=12,y  =  4,  c  =  2. 

39.  Evaluate  ^^±3^  when  a  =  3,  &  =  4,  c=6. 


2a -Qb 
3c 


40.  Evaluate  — — —  when  a=7,  b  =  2,  c=l. 


41.  Evaluate  '^  J^^\^-  when  o  =  13,  6=4,  c=13. 
2a— 0 


58  BEGINNERS'  ALGEBRA 

44.  General  equations.  Let  us  consider  a  number  of 
special  problems. 

(1)  What  nimiber  added  to  3  equals  5? 
The  algebraic  statement  is  3+^  =  5. 

(2)  What  number  added  to  2  equals  6  ? 
The  algebraic  statement  is  2-\-x  =  Q. 

(3)  What  niunber  added  to  7  equals  13  ? 
The  algebraic  statement  is  7+%  =  13. 

The  three  problems  are  of  the  same  type.  They  differ 
only  in  the  particular  known  numbers  used.  We  may  make 
the  statement  of  the  problem  so  general  that  it  will  include 
not  only  these  three,  but  all  problems  of  this  type  that  can 
be  thought  of. 

What  number  added  to  a  equals  b? 
The  algebraic  statement  is  a-{-x  =  b. 

We  call  this  a  general  equation.  Our  interest  is  centered 
on  one  unknown  which  we  represent  by  the  letter  x  so  that 
it  will  stand  out  clearly  from  the  other  numbers  of  the 
equations.  It  is  quite  common  to  use  one  of  the  last  letters 
of  the  alphabet  for  the  unknown  of  an  equation,  though 
this  is  not  at  all  necessary. 

Any  special  case  may  be  obtained  from  the  general  equa- 
tion by  the  insertion,  in  place  of  a  and  6,  of  particular 
numbers,  such  as  49  and  151: 

49+^=151 
A  question:  Can  a  and  b  be  any  numbers  whatsoever? 
Now  let  us  attend  to  the  solution  of  the  following  equa- 
tions : 

(1)  3+rx;  =  5  '  (2)  2-\-x  =  Q 

x=5-3  x=Q-2 

=  2  =4 

(3)  7-fic  =  13  (4)  a-\-x  =  b 

ic  =  13-7  x  =  b-a 

=  6 


THE  SOLUTION  OF  PROBLEMS  '     59 

All  four  are  solved  in  exactly  the  same  way.  In  each  case 
subtraction  is  used  to  get  the  known  ntimber  out  of  the  place 
where  it  is  not  wanted.  It  will  be  noticed  that  we  can 
obtain  the  answers  to  the  first  three  from  the  solution  of  the 
fourth  or  general  case  by  simply  putting  the  proper  numbers 
in  place  of  a  and  b. 

For  instance,  in  Exercise  2,  a  =  2,  6  =  6. 
The  general  solution  is 

x  =  b—a 

The  solution  of  Exercise  2  is 

^=6-2=4 
What  is.  the  solution  of  a  case  in  which  a  =  30  and  6  =  17? 
In  the  solution  of  the  general  case 
a-{-x=b 

every  example  of  the  kind  is  solved  once  for  all.  The  answer 
to  the  general  equation  may  be  used  as  a  formula  for  finding 
the  answer  for  any  partictdar  case.  We  may  call  the  a 
and  the  b  general  numbers.  Such  a  general  equation  is 
sometimes  called  a  literal  equation. 

Equations  in  which  all  the  nimibers  except  the  unknown 
are  figures  are  called  numerical  equations. 

The  numerical  equations  4;c  =  12,  6^  =  15,  7^  =  28  are 
special  cases  of  the  general  equation 

ax  =  b 
The  solution  of  the  general  equation  is 

b 

x  =  - 
a 

Using  this  result,  write  down  the  solution  for  each  of  the 
other  equations.  Notice  that  as  4  is  called  the  numerical 
coefficient  of  x  in  4^,  a  is  called  the  general  coefficient  of  x 
in  ax.     Ordinarily  we  simply  say  that  a  is  the  coefficient  of  x. 

In  working  with  general  numbers  we  use  them  just  as  we 


60  BEGINNERS'  ALGEBRA 

use  special  numbers  except  that  we  cannot  combine  them 
into  one  number  and  express  them  in  a  single  symbol: 

2+3  =  5 

but  a+6 

has  to  be  left  in  that  form. 

EXERCISES 

Solve  the  following  equations  for  x: 

1.  4a;+5  =  9  2.  2x-1  =  b  3.  2a;+ll  =  21 

4,  oa;+36  =  86  5.  ax-2h  =  ^h  6.  hx-Zc=l2c 

7.  cx-\-^a=lba  8.  ax-h  =  c  9.  hx-{-c=2h 

10.  cix; -30=56  11.  a+c:x:=3  12.  a-\-h  =  2x 

13.  o=c:)c+6  14.  o-fe+^«=0 

45.  Generalized  problems.     Consider  the  problem:    The 
sum  of  9  times  a  number  and  3  times  the  number  is  72. 

What  is  the  number  ? 

Let  n  be  the  number  and  the  statement  is 

9w+3w  =  72 

12w  =  72 

w  =  6 

or,  indicating  each  step,  9w+3w  =  72 

(9+3)w  =  72 

72 


w  = 


9+3 
6 


Let  us  state  a  problem  that  will  include  all  problems  of 
this  kind  as  special  cases.  The  sum  of  a  times  a  number 
and  h  times  the  number  is  c.    What  is  the  number  ? 

an-\-bn  =  c 

It  will  be  solved  in  exactly  the  same  way  as  the  special 

case  above: 

(a-\'b)n  =  c 

c 

a+6 


THE  SOLUTION  OF  PROBLEMS  61 

The  checking  of  this  problem  may  give  you  some  trouble. 
If  you  can  think  it  out,  weU  and  good;  if  not,  let  it  go  to  a 
later  time.  You  can  easily  check  yoiir  result  by  taking  a 
special  case.     For  instance,  take 

a=2,  6  =  3,  c=12,  or  a  =  l,  6=2,  ^=5 

Substitute  these  numbers  in  the  general  answer.  Then 
substitute  the  result  and  the  other  special  numbers  in  the 
original  equation.  Is  there  any  choice  between  the  two 
sets  of  nimibers  suggested  for  this  particular  example? 
Does  it  make  any  difference  what  values  are  assigned  to  the 
general  numbers  for  the  purpose  of  checking  results? 

It  is  to  be  noticed  that  such  expressions  as  ax-\-hx  can 
be  added  into  one  term  in  the  same  way  as  2^+3%,  by  the 
simple  addition  of  the  coefiBcients.  The  tmknown  number 
in  each  term  is  the  same. 

Special  case,  2x+dx=  {2+3)x  =  5x 

General  case,  ax-\-bx  =  {a-\-b)x 

In  the  special  case  the  coefficients  can  be  actually  added 
together  into  one  niunber.  In  the  general  case  the  addition 
can  be  expressed  only.  That  is,  this  addition  of  terms  can- 
not be  worked  out  with  a  case  like  ax+bx. 

MISCELLANEOUS   PROBLEMS 

State  as  equations  and  solve: 

1.  Three  times  a  number  is  two  less  than  17.  What  is  the  num- 
ber? 

2.  Three  times  a  number  is  b  less  than  17.    What  is  the  number? 

3.  Write  the  equation  called  for  in  Exercise  2  for  the  special 
cases  6  =  4,  7,  3,  5,  9.    What  is  the  required  number  in  each  case? 

4.  Three  times  a  number  is  b  less  than  c.    What  is  the  number? 

5.  State  the  special  cases  of  Exercise  4  if 

(1)  5=5,  c=12  (2)  &=3,  c=5 

(3)  J  =  6,  c=21  (4)  b='7,  c=19 

What  is  the  required  number  in  each  case? 


62  BEGINNERS'  ALGEBRA 

6.  a  times  a  number  is  b  less  than  c.    What  is  the  ntmiber? 

7.  Write  the  equation  called  for  in  Exercise  6  for  the  special  cases 

(1)  o  =  12,  6=10,  c=46 

(2)  o  =  5,  6  =  0,  c=17 

(3)  a=7,  b=S,  c  =  17 

(4)  o=9,  6  =  5,  c=32 

(5)  o=15,  6=5,  c  =  50 

Find  the  required  nimiber  in  each  case  by  using  the  solution  of 
the  general  equation  found  in  Exercise  6  as  a  formula. 

8.  Find  two  consecutive  integers  whose  sum  is  77;  whose  sum 
is  99. 

9.  Find  two  consecutive  integers  whose  sum  is  s. 

10.  Use  the  answer  found  in  Exercise  9  as  a  formula  and  find 
two  consecutive  integers  whose  sum  is  311,  401,  2395,  151,  28. 

11.  What  kind  of  a  number  must  s  be  in  Exercise  9  in  order  that 
the  problem  can  have  an  answer? 

12.  State  a  rule  for  finding  two  consecutive  integers  if  their  sum 
is  given. 

13.  The  perimeter  of  a  rectangular  field  is  p  feet;  one  side  is  a 
feet  longer  than  the  other  side.  What  is  the  width  of  the  field? 
The  length? 

14.  One  side  of  a  triangle  is  3  feet  longer  than  the  base;  the  other 
side  is  5  feet  longer  than  the  base.  What  is  the  length  of  the  base 
if  the  perimeter  is  32  feet? 

15.  Generalize  Problem  14. 

16.  The  angles  of  a  certain  triangle  are  such  that  the  second  is  a 
more  than  the  first  and  the  third  is  6  less  than  the  first.  What  are 
the  angles?    Apply  to  a  special  case. 

17.  In  a  certain  triangle  the  second  angle  is  a  times  the  first 
angle  and  the  third  angle  is  6  times  the  first.    What  are  the  angles? 

18.  Divide  any  number  into  two  parts  such  that  the  difference 
between  them  shall  be  n.  Apply  to  the  special  case  when  the 
ntmiber  is  241  and  n  =  5.  Can  you  use  any  niunbers  you  may 
choose  for  the  ntimber  and  for  n? 


THE  SOLUTION  OF  PROBLEMS  63 

19.  Divide  a  number  into  two  parts  such  that  one  part  shall  be 
n  times  the  other.  What  can  you  say  about  the  special  values 
that  can  be  given  to  the  number  and  to  n? 

20.  The  tens  figure  of  a  number  of  two  figures  is  twice  the  units 
figure.  What  are  the  figures  of  the  nimiber  if  their  simi  is  a? 
What  sort  of  a  number  must  a  be?    Why? 

21.  The  tens  figure  of  a  number  of  two  figures  is  n  times  the 
units  figure.  What  are  the  figures  if  their  sum  is  a?  Apply  to 
the  special  case  w  =  3,  a  =  12.  * 

46.  Finding  values  of  formulas.  The  results  obtained  by 
the  solution  of  generalized  problems  can  be  used  as  formulas 
for  finding  the  desired  answers  in  special  cases  without  its 
being  necessary  to  go  through  all  the  work  of  solving  the 
problem  again.  All  one  needs  to  do  with  such  a  formula  is 
to  put  in  the  special  numbers  and  perform  the  indicated 
arithmetical  work.  Such  formulas  are  of  great  use  to 
engineers,  mechanics,  manufacturers,  and  many  others. 
Important  formulas  have  been  worked  out  by  men  well 
acquainted  with  the  kind  of  work  in  which  they  are  to  be 
used  and  then  preserved  in  handbooks  for  the  use  of  those 
who  may  need  them.  The  ability  to  use  such  formulas  and 
get  the  desired  results  from  them  is  of  great  importance. 
Such  formulas  are  shorthand  rules  for  calculating  certain 
desired  numbers. 

EXERCISES 

Substitute  in  the  following  formulas  and  make  the  necessary 
arithmetical  calculations. 

Note.  All  of  these  formulas  are  of  use  in  practical  work.  They 
are  taken  from  various  trades  and  professions. 

1.  A=^,  a=12,  b  =  2Q 

2-  ^=^^^5^>  M  =  30,  A^  =  24,  P=14 

3.  PF  =  ^,  P  =  45,  j  =  5,  ^  =  3 

a 


64  BEGINNERS'  ALGEBRA 


4.  W=^,  P=36,  5«2i,  d^l2 
a 

5    c-^    P-i 


6.  TF«7.6Z)-1.5,  Z?  =  j 

8.  A=Ji.+  .01,  «=5 

2n 

9.  R=D-^^,  Z>=2,  r=12 

10.  3="^,  a=5,  r=3,  /=125 

f  —  1 

11.  r=212-A    A  =  7000 

12.  H=20v-,  /  =  3,  «=15,  t;=1320 

/J 

13.  y=2X^,  a  =  1.175,  5  =  2.578 

14.  Tr=P-?^,  P=223,  R=19,  a=12,  ^=4 

a— 0 

15.  ir  =  ^+32,  C=30 

5 

16.  /=a+(»-l)£/,  a  =  3,  w==10,  (f=4 

17.  5=^(a+/),  »  =  100,  a=l,  /  =  100 


CHAPTER  IV 
Negative  Numbers 

47.  A  difficulty.     Consider  the  general  equation 

x—b  =  a 
Its  solution  is  x  =  a+b 

No  matter  what  values  are  given  to  a  and  b,  their  sum  can 
always  be  found.     Every  equation  of  this  kind  can  be  solved. 

But  consider  the  general  equation 
x+b  =  a 


The  solution  is 

x  =  a—b 

For  the  special  case 

a  =  9,  6  =  5 

this  is 

i^  =  9-5 

=  4 

But  for  the  special  case 

a  =  9,  6  =  12 

we  have 

^  =  9-12 

Here  we  meet  a  difficulty,  for  we  have  learned  in  arithmetic 
that  we  cannot  subtract  a  larger  number  from  a  smaller  one, 
and  we  are  led  to  say  that  the  equation  x-\-b  =  a  cannot  be 
solved  in  all  cases,  but  only  in  those  special  cases  when  b  is 
not  greater  than  a.  And  so  man  thought  for  many  years, 
tmtil  the  Hindus  of  the  fifth  or  sixth  century,  who  were  not 
satisfied  to  let  the  matter  rest  there,  invented  a  way  of 
overcoming  the  difficulty. 

48.  Some  practical  illustrations.  It  is  worth  while  to 
examine  a  little  more  carefuUy  the  statement  made  in  the 
last  article  that  '*we  cannot  subtract  a  larger  number  from 
a  smaller."  It  is  true  that  it  is  impossible  to  take  5  apples 
from  a  plate  on  which  there  are  but  3.  But  one  can  easily 
determine  the  resulting  temperature  when  15°  have  been 

65 


66  BEGINNERS'  ALGEBRA 

taken  from  the  temperature  of  a  room  in  which  the  tempera- 
ture is  registered  as  10°.  There  has  been  a  drop  of  15°; 
15°  have  been  taken  away.  We  say  the  result  is  5°  below 
zero.  Five  degrees  above  zero  and  5°  below  zero  have  very 
definite  meanings. 

Or  again,  it  is  easy  to  determine  the  result  of  buying  a 
coat  costing  $25  when  one  has  only  $15.  A  person  cannot 
take  $25  from  his  pocket  when  there  is  but  $15  in  it,  but 
he  can  buy  the  coat  by  going  in  debt  for  $10. 

A  man  with  only  $3000  can  buy  a  house  for  $5000  by 
paying  the  $3000  and  going  in  debt  for  the  remaining  $2000. 
He  takes  $5000  from  his  assets,  which  leaves  him  a  debt  of 
$2000. 

Assets  and  debts,  temperature  above  zero  and  temperature 
below  zero  are  certainly  different  kinds  of  numbers. 

49.  Usefulness  of  negative  numbers.  It  is  convenient  to 
have  some  way  to  distinguish  clearly  the  one  kind  of  number 
from  the  other.  To  meet  this  need  a  new  kind  of  number  has 
been  invented.  We  will  use  the  numbers  already  invented, 
such  as  3,  7,  |,  for  one  kind  of  quantity,  such  as  assets, 
above  zero,  etc.,  and  call  them  positive  numbers,  and 
denote  them  by  using  the  sign  +  in  front  of  them:  +25. 
We  will  invent  a  new  kind  of  nimiber  to  represent  the  other 
quantities,  such  as  debt,  below  zero,  and  call  them  negative 
numbers,  using  the  same  figures  but  placing  the  sign  —  in 
front  of  them:  —25.  Thus  +75°  means  75°  above  zero, 
and  —75°  means  75°  below  zero. 

Note.  The  origin  of  the  signs  +  and  —  is  not  quite  certain,  but  it 
is  probable  that  some  mathematician  saw  them  on  the  chests  of  goods 
in  a  German  warehouse  about  five  hundred  years  ago.  They  were 
used  to  indicate  whether  the  weight  of  a  given  chest  was  over  or 
under  a  given  standard,  weight.  They  appear  for  the  first  time  in  a 
mathematics  book  in  one  written  by  Widmann  in  1489. 

The  plus  sign  (+)  may  be  omitted  in  the  case  of  positive 
numbers. 


NEGATIVE  NUMBERS  67 

EXERCISES 

1.  If  +$25  represents  a  credit,  how  would  you  represent  a  debt 
of  $10? 

2.  If  53  represents  a  gain,  what  does  —47  denote? 

3.  In  a  certain  dictionary  you  will  find  the  dates  of  the  births  of 
George  Washington  and  Julius  Caesar  put  down  thus :  1732,  — 100 . 
What  do  the  numbers  mean? 

4.  In  the  same  dictionary  you  will  find  the  elevations,  Mt.  Vesu- 
vius 3948  feet,  Dead  Sea  1312  feet  below  Mediterranean  Sea. 
Express  these  elevations  by  using  positive  and  negative  numbers. 
Elevation  means  the  vertical  distance  from  sea  level. 

5.  How  woiild  you  express  in  numbers  the  latitudes  of  New  York 
and  Rio  de  Janeiro? 

6.  Express  in  algebraic  numbers  10  miles  west  and  15  miles  east; 
the  freezing  point  of  water  and  the  lowest  temperature  reached  in 
the  place  where  you  live;  also  the  boiling  point  of  water. 

7.  Express  in  algebraic  symbols  the  speed  of  a  train  going  ahead 
and  the  speed  of  a  train  backing;  the  weight  of  a  wagon  and  the 
weight  of  a  balloon  which  is  ready  to  be  cut  loose;  underweight 
and  overweight. 

50.  Number  system.  With  the  addition  of  negative  num- 
bers we  have  now  three  kinds  of  numbers  that  can  be  used 
when  needed — integers,  fractions,  including  both  common 
and  decimal,  and  negative  numbers.  Integers  and  fractions 
have  been  used  a  very  long  time.  We  do  not  know  when 
they  were  first  invented.  The  early  Egyptians  used  frac- 
tions with  unit  nimierators,  J,  ^,  -s^,  etc.  These  are  found 
in  the  most  ancient  book  on  arithmetic,  written  by  an 
Egyptian  named  Ahmes.  The  fractions  used  by  the  Baby- 
lonians, as  showTi  on  tablets  that  have  been  unearthed,  had 
the  same  denominator,  60  being  used  for  that  piupose. 
For  instance,  ^  was  represented  by  the  s3mibol  for  30,  the 
word  "sixtieths"  being  left  for  the  reader  to  supply.  The 
Greeks  used  fractions  of  a  more  general  type,  such  as  ^. 
None  of  these  ancient  peoples  used  decimal  fractions  or 


68  BEGINNERS*  ALGEBRA 

negative  numbers.  Strange  as  it  may  seem,  negative 
numbers  were  invented  before  decimal  fractions.  It  is 
probable  that  negative  numbers  are  due  to  the  Hindus, 
who  were  very  skillful  in  arithmetic  and  algebra.  Aryab- 
hatta,  who  was  bom  in  476  a.d.,  distinguished  between 
positive  and  negative  numbers,  but  without  the  signs  plus 
and  minus. 

The  decimal  fraction  is  a  comparatively  recent  innovation. 
The  idea  was  of  slow  growth,  but  was  used  effectively  toward 
the  close  of  the  sixteenth  century  by  Simon  Stevin  of  Bruges, 
Belgium,  and  others  in  making  certain  calculations.  It 
was  not  until  the  beginning  of  the  nineteenth  century  that 
decimals  were  brought  into  ordinary  arithmetic.  Various 
notations  have  been  used  for  fractions.  The  Hindus  wrote 
a  fraction  in  the  form  of  § ;  the  bar  we  use  between  the  two 
parts  of  a  fraction  f  was  probably  introduced  by  the  Arabs. 
One  of  the  forms  Stevin  used  for  a  decimal  was  5  (0)  9  (1)  1 
(2)  2  (3);  our  form  is  5.912. 

51.  Number  scale.  You  know  how  the  numbers 
are  arranged  on  a  thermometer.  Each  mark  indi- 
cates a  temperature  lower  than  the  mark  above  it. 
(Fig.  7.)  Usually  on  the  thermometers  we  use  at 
our  houses  the  change  from  one  mark  to  another  is 
the  change  of  one  degree. 

Such  a  scale  furnishes  a  very  convenient  way  of 
arranging  positive  and  negative  njimibers,  the  integers  lo 
being  placed  at  equal  distances  along  a  straight  line 
in  such  a  way  as  to  show  their  distance  from  a  point 
chosen  as  the  starting,  or  zero  point.  It  is  customary 
to  put  the  positive  numbers  on  the  right .  Each  number  repre- 
sents not  only  a  numerical  value,  but  also  that  numerical 



-9-8-7-6-5-4-3-2-1      0       1       2      3      i       5      «      7      8      » 

Fig.  8 
value  in  a  given  direction  from  the  zero  point.    Each  integer 
is  one  less  than  the  next  integer  to  its  right ;  each  is  less  than 
any  integer  to  its  right:  2  is  less  than  5,  —4  is  less  than  —2. 


NEGATIVE  NUMBERS  69 

EXERCISES 

1.  Where  on  the  scale  would  you  place  +12,  —15;  —11,  —i, 
-i,  -2i,  +f,  -h  .5,  -2.2,  -3.5? 

2.  What  is  the  number  that  is  halfway  between  5  and  6?  Half- 
way between  —5  and  —6? 

52.  Adding  a  positive  number.  We  may  think  of  passing 
from  one  point  on  the  scale  to  the  next  point  on  the  right 
as  coimting  on  one  or  adding  one  positive  unit. 

To  add  3  to  7,  we  may  start  at  7  on  the  scale  and  count 
on  3  units  to  the  right  and  reach  10.  To  add  3  to  —7,  start 
at  —7  and  count  on  3  units  to  the  right  and  reach  —4. 

EXERCISES 

By  means  of  the  scale  add: 

1.  5  to  9,  6  to  3,  3  to  6,  2  to  -3 

2.  5  to  -7,  9  to  -3,  5  to  -5,  7  to  -2 

3.  10  to  -7,  7  to  -9,  3  to  0,  3  to  5 

53.  Subtracting  a  positive  number.  In  a  similar  way  we 
may  think  of  passing  from  right  to  left  as  counting  off  or 
subtracting  a  unit  at  each  step.  Subtraction  is  the  undoing 
of  addition.  Therefore  in  subtraction  we  reverse  the  direc- 
tion and  pass  to  the  left. 

To  subtract  3  from  7,  we  may  start  at  7  and  cotmt  off  3 
to  the  left  and  reach  4.  To  subtract  3  from  —5,  start  at  —5 
and  coimt  off  3  to  the  left  and  reach  —8. 

EXERCISE  I 

By  the  use  of  the  scale  subtract: 

1.  2  from  7,  7  from  7,  9  from  7 

2.  5  from  2,  3  from  5,  8  from  1 

3.  5  from  5,  5  from  0,  2  from  —3 

4.  3  from  -2,  4  from  -1,  2  from  -6 

We  may  write  such  examples  in  mathematical  symbols, 


70 


BEGINNERS'  ALGEBRA 


using  the  distinguishing  marks  for  positive  and  negative 
numbers. 

Add  2  to  -3,  (-3)  +  (+2) 

Subtract  3  from  (-5),  (-5) -(+3) 
or,  omitting  the  sign  for  the  positive  number, 

(-3)+2 
(-5)-3 

since  3,  +3,  and  (+3)  all  mean  the  same  thing. 


EXERCISE   II 

Find  values  of  the  following  from  the  number  scale: 


1.  (-3)  +  (+5) 

3.  (-7)  +  (+2) 

5.  9-8 

7.  (-9)+3 

9.  (5)-(4-6)+(+2) 
11.  7-6+8 
13.  2-5+6 
15.3-9+10 
17.  7+2-3 
19.  (-2) +  (3) -(+7) 
21.  (-9)+6-l 
23.  8-5+3 


2.  (6) +  (+3) 

4.  (-7)  +  (+9) 

6.  9-12 

8.  (-9) -2 
10.  9-5+3 
12.  3-(+7)  +  (+l) 
14.  2-7+4 
16.  7 -(+2) -(+3) 
18..  (-2)+7-3 
20.  (-2) +5 -6 
22.  (-9)+9-2 
24.  8-12+2 


25.  If  the  temperature  is  now  +10°,  what  will  represent  the 
temperature  after  a  fall  of  8°?    After  a  fall  of  15°? 

26.  If  the  temperature  is  now  —5°,  what  will  represent  the 
temperature  after  a  rise  of  8°?    After  a  fall  of  8°? 

27.  A  boy  has  $3  and  buys  some  tools  worth  $5.  What  will 
represent  the  condition  of  his  finances? 

28.  While  playing  a  certain  game  a  man  finds  himself  "3  in  the 
hole."  What  will  represent  his  score  if  he  gains  8  points?  If  he 
loses  2  points? 


NEGATIVE  NUMBERS  71 

29.  Jerusalem  is  2500  feet  above  sea  level.  To  reach  the  Dead 
Sea,  one  descends  3800  feet.  What  number  represents  the  level 
of  the  Dead  Sea? 

30.  A  boy  is  15  pounds  overweight.  He  loses  20  pounds.  What 
number  represents  his  condition? 

54.  Solution  of  equations.  In  the  last  article  it  was  shown 
that  the  invention  of  negative  numbers  enables  us  to  subtract 
a  larger  number  from  a  smaller.  It  therefore  enables  us  to 
solve  the  equation  that  started  this  discussion: 

^+9  =  5 
^  =  5-9 
=  -4 
Check:  (-4) +9  =  5 

Use  the  number  scale  in  finding  value  of  the  left  side. 

EXERCISES 

Solve  and  check: 

1.  x-\-9  =  8  2.  x+l2  =  4:  3.  a:+13  =  4 

4.  a:+15  =  12  5.  x-\-n  =  p  6.  ^+7=1 

7.  Derive  equations  from  Exercise  5  in  which 

(1)  «=12,  p  =  Q  (2)  »  =  9,    p  =  2 

(3)  w=13,  p  =  3  '  (4)  «  =  15,  p  =  S 

8.  Find  the  answers  to  the  equations  obtained  in  Exercise  7  by- 
substituting  the  numbers  given  in  the  answer  to  Exercise  5. 

55.  Adding  on  the  scale.  Adding  a  positive  number 
means  counting  to  the  right.  Adding  a  negative  number 
means  counting  to  the  left. 


-9-8-7-6-5-4-3-2-1      0       1       2      3      4       5      «      7      8      » 

Fig.  9 

Add  3  to  5,  start  at  5,  move  3  places  to  right,  reach  8. 

Add  —3  to  5,  start  at  5,  move  3  places  to  left,  reach  2. 

Add  7  to  —2,  start  at  —2,  move  7  places  to  right,  reach  5. 

Add  —5  to  —2,  start  at  —5,  move  2  places  to  left,  reach 

•7. 

Add  2  to  —5,  start  at  ?,  move  ?  places  to  ?,  reach  ? 


72  BEGINNERS'  ALGEBRA 

EXERCISES 

Find  the  value  of  the  following  from  the  number  scale: 

1.  (6)  +  (-8)  2.  (8)+(-12) 

3.  (-2)  +  (-9)  4.  (-.6)+(+3) 

5.  (7)+ -10  6.  7+(-3) 

7.  (-7)+(-10)  8.  (-9)+(-3) 

9.  6+(-4)+2  10.  7+(-10)+3 

11.  7+(-10)  +  (-3)  12.  (-3)+6+(-8) 

13.  (-3)  +  (-6)4-8  14.  (_3)  +  (-6)+(-8) 

15.  (+3)  +  (-6)+8  16.  3+(-6)  +  (-8) 

56.  Addition.     The  way  to  proceed  is  made  clear  by  the 
special  problem  of  combining  debts  and  credits. 

The  result  of  adding  a  credit  to  a  credit  is  clearly  a  credit, 
equal  to  the  sum  of  the  two  amounts  of  money: 

+25 

+30 

+55 

The  result  of  adding  a  debt  to  another  debt  is  clearly  a 

debt,  equal  in  value  to  the  sum  of  the  two  amounts  of  money: 

-25 

-30 

-55 

There  is  no  difficulty  in  finding  the  result  of  combining  a 

debt  and  a  credit.     It  will  be  a  credit  if  the  amount  of  the 

given  credit  is  greater  than  the  amount  of  the  given  debt; 

and  a  debt  if  the  amount  of  the  given  debt  is  greater  than 

the  amount  of  the  given  credit: 

.   +25  +25 

-15  -30 

+  10  -  5 

In  each  case  the  amount  of  the  result  is  the  difference 
between  the  given  amounts. 


NEGATIVE  NUMBERS  73 

The  illustration  suggests  an  extension  of  the  idea  of  adding. 
In  each  case  the  term  "adding"  has  been  used.  When  the 
two  numbers  were  both  positive  or  both  negative^  the  adding 
impHed  the  finding  of  an  arithmetical  sum.  When  one 
niimber  was  positive  and  the  other  negative,  the  adding 
implied  the  finding  of  an  arithmetical  difference.  Both 
operations  are  included  in  what  we  call  addition. 

In  arithmetic  the  stmi  of  two  numbers  is  always  greater 
than  either  ntimber.  This  is  not  always  true  in  algebra. 
The  sum  of  two  numbers  may  be  less  than  either  or  less 
than  both  the  numbers.  Show  that  this  last  statement  is 
true. 

57.  Definitions.  These  illustrations  suggest  certain  defi- 
nitions. The  numerical  value  of  a  nimiber  (disregarding  its 
sign)  is  called  its  absolute  value. 

The  sum  of  two  positive  numbers  is  a  positive  number 
whose  absolute  value  is  the  sum  of  the  absolute  values  of 
the  numbers. 

The  sum  of  two  negative  numbers  is  a  negative  nimibei 
whose  absolute  value  is  the  stmi  of  the  absolute  values  of  the 
two  numbers. 

The  simi  of  a  positive  and  a  negative  ntmiber,  or  of  a 

negative  number  and  a  positive  number,  is  a  positive  or  a 
negative  number  according  as  the  positive  or  the  negative 
number  is  in  excess.  Its  absolute  value  is  the  difference 
between  the  absolute  values  of  the  two  numbers. 

58.  Rule  for  addition.  We  may  write  a  practical  rule  for 
addition  in  this  simple  form: 

(a)  If  the  numbers  have  the  same  sign,  find  the  stmi  of 
their  absolute  values  and  prefix  the  common  sign. 

(6)  If  the  numbers  have  different  signs,  find  the  difference 
between  the  absolute  values  and  prefix  the  sign  of  the  one 
having  the  larger  absolute  value. 

6 


74  BEGINNERS'  ALGEBRA 


EXERCISES 

Add: 

1.      30    5     -9     -3 

-9       -9        25     -6    19 

3.8 

-10    8        4-7 

+3     +15     -25     -2    23 

-2.9 

2.      2        3a          9a 

-23jc     -25n     -Qn     -2 

3     -2a     -11a 

-  2x     +30n        6n     +3 

3.  Find  the  sum  of  the  following  credits  and  debits: 
+25,  +10,   -15,  +3,  -10,   -7 

We  may  find  the  sum  of  several  numbers  by  adding  each 
one  in  succession.  A  preferable  method  is  to  find  the  sum  of 
the  positive  numbers  and  the  sum  of  the  negative  numbers, 
and  then  find  the  simi  of  these  two  results.  There  is  less 
chance  of  making  a  mistake  when  the  latter  plan  is  followed. 

4.  Find  the  sum  of: 

15,  -7,  +6,  3,   -5,  +2,   -8 

5.  Add: 

-7,  9,   -3,  2,  11,   -12,  +3,   -10 

6.  Find  the  sum  of: 

3a,   -5a,   -2a,  +8a,   -10a 

7.  Add  together: 

Qx,   -2x,  25:»,    -15x,  28:v,    -4:X 

8.  Find  the  sum  of: 

7n,  2»,  —3n,  — 4w,  +5w 

9.  Find  the  sum  of: 

2y,    -5^,  +3^,    -2y,  -\-y,   -7y 

10.  Add  together: 

15:r,   -7x,   -2x,  -\-4cX,   -lOa; 

11.  Find  the  sum  of : 

—3a,  —2a,  6a,  7a,   —4a,  5a 

12.  Find  the  average  of: 
6,  7,  6,  9,  7,  10,  6,  5 

13.  Find  the  average  of: 
65,  80,  0,  75 


NEGATIVE  NUMBERS  75 

14.  Find  the  average  of: 
6,   -2,  8,   -5,  3 

15.  What  is  the  average  temperature  when  the  readings  taken 
each  hour  are  28,  12,  8,  4,  0,  -2,  -5,  -7,  -6,  0,  5,  5? 

16.  Let  the  members  of  the  class  guess  the  length  of  a  line  drawn 
on  the  blackboard.  The  teacher  will  measure  the  line  and  announce 
the  length.  Let  each  student  subtract  the  true  length  from  his 
guess  and  announce  his  result  to  be  written  on  the  blackboard. 
Then  let  each  member  of  the  class  compute  the  average  of  all  these 
errors. 

59.  Double  meaning  of  signs  +,  — .  We  have  used  the 
expression  (+7) +  (  —  3)  to  mean  the  sum  of  positive  7  and 
negative  3,  the  signs  within  the  parentheses  marking  the 
distinction  between  positive  and  negative.  And  we  have 
also  used  a  simpler  expression  for  positive  numbers,  simply 
7  without  any  sign:  7+ (  —  3). 

In  adding  —3  to  7  on  the  number  scale  we  move  three 
places  to  the  left  and  reach  4,  which  is  the  same  operation 
as  subtracting  3  from  7,  and  the  same  result  is  obtained. 
We  may  say  then  that  adding  a  negative  number  gives  the 
same  result  as  subtracting  a  positive  number  of  the  same 
absolute  value. 

We  may  then  replace 

7+(-3)by7-3 

so  that  7  —  3  may  be  thought  of  in  two  ways:  as  meaning 
7  minus  3 ;  or  as  meaning  the  sum  of  positive  7  and  negative 
3,  or,  as  we  often  speak  of  it,  the  sum  of  +7  and  —3. 

It  is  the  usual  practice  to  regard  7x—3x  as  the  sum  of 
the  two  terms  or  nimibers : 

7x  and  —3x 
Thus  3+7a-5 

is  thought  of  as  the  sum  of 

3,  7a,  and  —5 


76  BEGINNERS'  ALGEBRA 

EXERCISES 

Write  each  of  the  following  expressions  in  another  way,  explain 
the  meaning  of  each,  and  find  results: 

1.  6+(-3)  2.  6-3  3.  2+(-5) 

4.  2-5  5.  7+ (-5)  6.  8-2 

7.  9-11  8.  7a+(-8a)  9.  5x-7x 

10.  3a-12o  11.  2o+(-9a)  12.  7a+(-3a) 

60.  To  find  the  sum  of  several  positive  and  negative 
numbers.  It  is  a  simple  matter  to  express  the  sum  of 
several  positive  and  negative  numbers.  To  find  the  sum  of 
X,  —a,  -\-b,  —3,  write  down  each  number  in  order  with  its 
sign  in  front : 

+x-a+b-S 

The  sum  of  —2,  4x,  —5,  —2x,  is  —2+4x  —  5  —  2x  or 
2:31; -7.  If  the  first  nimiber  in  the  series  is  a  positive 
number,  the  sign  need  not  be  written.  The  sum  of  x^ 
—  2,  —3a,  7,  is  x—2  —  Sa+7.  We  usually  regard  the  sign  in 
front  of  a  term  as  belonging  to  the  term  and  speak  of 
positive  and  negative  terms.  In  7it;— 3+4^— 10^  the  terms 
are  +7x,  —3,  +ix,  —10:^;. 

EXERCISES 
Find  the  sum  of  the  following: 

1.  3,  -2,  5,  -7,  -9,  +3 

2.  3o,  -7a,  4-2a,  +5a,  -a,  -9a 

3.  9x,  -X,  +7x,  -5x,  4-10:*:,  -Sx 

4.  SA,  A,  25A,  -SA,  -4U,  +31^,  6^4 

5.  47r,  -67r,  +37r,  -27r,  -97r 

6.  57r-37r+27r-97r+77r 

7.  5+5w-7+2w-8w+9+7«-2 

8.  7,  -3:^,  -9,  -a,  +11,  -10a,  15a,  -10a; 

9.  7,  X,  -a,  -b,  +3a,  -2x,  +7x,  +36 


NEGATIVE  NUMBERS  77 

10.  2:r,   -f,  +1^,    -4,  +1,    -f:x;,  i,   -^x,   -Joe 

11.  3/-5+2/-7/+9+ll^ 

12.  3x-2,  2x+5,  5 -3a;,  2jc-5 

13.  3-y,  4+2)^,  7+5)',   -3-4y 

4 

61.  Adding  terms  of  different  kinds.  When  a  number 
of  expressions  with  several  kinds  of  terms  are  to  be  added 
together,  it  is  often  convenient  to  arrange  them  in  columns 
with  like  terms  under  each  other  just  as  we  do  in  arithmetic. 
This  makes  the  addition  somewhat  easier. 

345  )  I  300+40+5 

26  (  ...  )  20+6 

605  which  means  ^^^         _^^ 


976  I  (  900+60+16 

7x+Sa-7y+5b 
-a  ■^2y-Sb 
x—5a  —  Sy 
Sx-Sa-Sy-\-2b 

EXERCISES 

Find  the  sums  of  the  following: 

1.  4.2,  32.97 

2.  4.2,   -3.46 

3.  4.31,   -5.37 

4.  4,   -6.23 

5.  3.24,   -7.41,  +3.46 

6.  3,  6.84,  7.6,   -6.7,   -.98 

7.  2a -46,  Qa-\-5b,  2b -3a,  Sb-a 

8.  3a -26,  56 -3c,  2c -3a 

9.  ajc— 3^,  2ax—5y,  7y—6ax 

10.  3ic+2a+36,  2x-5a+26,   -5x-2a-96,  3:x+a+6 

11.  0+6— c,  a— 6+c,   — a+6+c,  a+6+c 


78  BEGINNERS'  ALGEBRA 

12.  ia-^x,  ia-^x,  ia+^x 

13.  Sax-2by,   -7ax-^9by,   -2ax-Sby 

14.  15:«-9,  -10^+3,  -47a;-31,+19x+3,  -7jc-21,  -dSx-\-35 

15.  7ic+41a -3,  35a -23:^+91,  72 -8ic-71<z,  -13+16a-25:»; 

16.  7x -3b  -2a,  7y-\-3a  -  10b,  136  -5a  -2y  -Sx,  Ua  -10y-6b-x 

17.  Sx—2a—b,  ia-^b-^x,  ^b—^a  —  ^x 

18.  5x-Qy-7z,  5y4-92;-4x,  2x-Sy-z,  y+4:Z-Qx 

19.  Sa-2b-\-4:C,  7a-\-Sb  -5c,  2c  -2a  -U,.  a+b  -6c 

20.  1.6jc-1.4y-z,  1.2:«;+1.33;+ .5z,  x-2y-\-l.Qz 

21.  9x-10y-9z,  lly+3z-2jc,  4:Z-5x-{-Qy,  x-7y-8z 

22.  %x-3y-z,  2x-^y-\z,  lx-y+2z 

23.  7x-d>y-\-l0a,  9y-5a-3x,  4:a-2y-Qx,  3:^-93'+2a,  7>'-6jc 

-9a 

24.  .7x-.5y-.Qz,   .7y-.5z-.6x,   .5x+ . 73^+1.52 

25.  2X-33;,  43;-5:r,  6^-73;,  5y-Qx,  3x+9>',  -23;-llx 

26.  a-b,  26 -4c,  c-3a,  3a -56 -2c,  66 -4a,  5a+c 

27.  ix-iy+ix-\-iy-ix-%y 

28.  2n-{-3h,  5n-7h-\-2,   -3n-h-5 

62.  Subtraction.  We  have  seen  that  the  adding  of  a 
negative  number  and  the  subtracting  of  a  positive  number 
of  the  same  absolute  value  had  the  same  result.  That  is, 
the  subtraction  of  a  positive  number  may  be  replaced  by 
the  addition  of  a  negative  number. 

The  question  then  arises,  how  shall  we  subtract  a  negative 
number?  Subtracting  is  the  reversal  of  adding.  If  adding 
a  negative  number  means  counting  to  the  left  on  the  num- 
ber scale,  then  subtracting  a  negative  number  must  mean 
counting  to  the  right,  which  is  the  same  operation  as 
adding  a  positive  number. 


-9-8-7-6-5-4-3-2-1    0    I    2    3    4    5    «    7    8    9 
Fig.  10 


NEGATIVE  NUMBERS  79 

Subtract  3  from  5 
Subtract  3  from  —5 
Subtract  —3  from  5 
Subtract  —3  from  —5 
Subtract  —2  from  7 
Subtract  —  7  from  2 
Subtract  —8  from  3 

We  will  assume,  then,  that  in  every  case  the  operation  of 
subtraction  can  be  replaced  by  addition.  There  is  nothing 
new  or  strange  in  this  idea  of  replacing  subtraction  by  addi- 
tion. It  is  used  constantly  in  making  change.  A  five-dollar 
bill  is  offered  in  payment  of  a  bill  of  $2.87.  How  much 
money  is  to  be  paid  back?  The  man  in  the  store  does  not 
find  the  amount  to  be  returned  by  subtraction,  5 .  00  —  2 .  87  = 
2.13,  but  by  addition,  2.87+ .03+.  10+2. 00. 

The  operation  of  subtracting  a  number  may  then  be 
expressed  in  the  form  of  a  rule : 

Rule.  To  subtract  a  positive  number,  add  a  negative 
number  of  the  same  absolute  value. 

To  subtract  a  negative  number,  add  a  positive  nimiber 
of  the  same  absolute  value. 

To  subtract  4  from  7,  add  -4  to  7,  7-4  =  3. 

To  subtract  4  from  3,  add  -4  to  3,  3-4=  - 1. 

To  subtract  -5  from  3,  add  5  to  3,  3+5  =  8. 

To  subtract  -5  from  -7,  add  5  to  -7,  -7+5=  -2. 

Consider  a  practical  case.  If  the  temperature  at  6 :  00  a.m. 
is  30°  and  at  noon  is  70°,  we  find  the  rise  in  temperature  by 
subtracting  30°  from  70°. 

70-30  =  40 

If  the  temperature  at  6:00  a.m.  is  —10°  and  at  noon  is 
40°,  we  find  the  rise  in  temperature  by  subtracting  —10 


80 


BEGINNERS'  ALGEBRA 


from  40,  that  is,  by  adding  + 10  to  40,  which  gives  50.    These 
illustrations  may  be  expressed  in  algebraic  symbols: 

(7)-(4)  =  7-4  =  3 
(3)-(4)=3-4=-l 
(3)-(-5)=3+5  =  8 
(-7)-(-5)  =  -7+5=-2 
(40) -(-10)  =40+10  =  50 


EXERCISE  I 

1.  Subtract  17  from  23,  42  from  23,  38  from  9,  -12  from  30, 
-20  from  16,  3  from  -9,  -6  from  -10,  -20  from  9,  -9  from  20, 
-10  from  -6. 

2.  From  5*  subtract  2Xj  from  Ix  subtract  —9a;,  from  —2a;  sub- 
tract —ZXf  from  —12a;  subtract  —^x. 

3.  Copy  and  fill  in  the  blanks  with  the  values  of  a— 6,  where  a 
represents  the  numbers  in  the  colimin  at  the  left  and  h  represents 
the  ntmibers  in  the  row  at  the  top. 


7 

-13 

3 

-5 

-9 

4 

1 

12 

-6 

10 

9 

-6 

7 

-10 

1 

-3 

4.  Copy  and  fill  in  the  blanks  in  the  table  at  the  top  of  page 
81  with  the  values  of  a— 6,  where  a  represents  the  numbers  in 
the  coltmin  at  the  left  and  h  represents  the  nimibers  in  the  row 
at  the  top. 


NEGATIVE  NUMBERS 


81 


3x 

-2x 

-5x 

-7x 

4w 

-3« 

9/ 

-12^ 

Tx 

-bx 

6x 

X 

3n 

-In 

-5/ 

+7/ 

EXERCISE  II 


Find  the  value  of  the  following: 


1.  3-(-6) 

3.  7+ (-2) 

5.  7-(-9) 

7.  4a; -(-3a:) 

9.  (-2) -(4) 
11.  (-4)  +  (-5) 
13.  6 -(2) +  (-3) 
15.  7 -(5) -(-9) 
17.  (_8)  +  (-6)-(-4) 
19.  (-3)-(-5)  +  (-5) 
21.  (2) -(4) -(-6) 
23.  (-2) +  (-4) -(+6) 


2.  4+(-6) 

4.  7 -(-2) 

6.  7+ (-9) 

8.  bx-^-i-Sx) 
10.  (-2) -(-4) 
12.  {-Sx)-^{-2x) 
14.  3-(-2)  +  (-3) 
16.  (_5)-(-7)  +  (10) 
18.  (10) +  (-6) -(-3) 
20.  (-3)  +  (-6)-(-7) 
22.  (-2) -(-4) +  (-6) 
24.  (+2)-(-4)  +  (-6) 


25.  If  a=5  and  b=6,  evaluate  a—b,  2a— 36. 


82  BEGINNERS'  ALGEBRA 

26.  If  a=  -2  and  6  =  7,  evaluate  a+&,  a-b,  a -36,  a+3o. 

27.  If  a  = -2,  b=-Q,  c=4,  evaluate  a+fc,  a-6,  a+b-c, 
a—b-\-c,  b—c—a,  b-\-c—a. 

28.  If  x=  -3,  y^2,  evaluate  x-5,  x+y+5,  y-x-\-7,'Sy-x-5. 

63.  Positive  and  negative  numbers.  Answers  must  be 
given  in  algebraic  language. 

EXERCISES 

1.  Add  7,  subtract  10,  add  —3,  subtract  —5,  add  9,  subtract  5, 
add  6,  add  —10,  subtract  -14,  subtract  16,  add  25,  subtract  —8. 

2.  Take  6a,  add  7a,  subtract  —9a,  subtract  llo,  add  —6a,  add 
9a,  subtract  a,  subtract  —3a,  add  5a,  add  —10a. 

3.  Frangois  Vieta,  the  first  man  to  write  ab  for  aXb,  was  bom 
in  1540  and  died  in  1603.     How  old  was  he  when  he  died? 

4.  Augustus  Caesar  was  bom  in  63  b.  c.  and  died  in  14  a.  d. 
How  old  was  he  when  he  died? 

5.  Pythagoras  was  bom  in  580  B.C.  and  died  in  501  b.  c.  How 
old  was  he  when  he  died? 

6.  At  3:00  P.M.  the  temperature  was  93°;  at  6:00  p.m.  it  was  75°. 
How  much  of  a  drop  was  there  in  temperature? 

7.  What  was  the  drop  in  temperature  from  10°  at  noon  to  15° 
below  zero  at  6:00  p.m.? 

8.  What  is  the  difference  in  latitude  between  New  York,  40°  40', 
and  Baltimore,  39°  187 

9.  What  is  the  difference  in  latitude  between  New  York,  40°  40', 
and  Rio  de  Janeiro,  23°  south? 

10.  What  is  the  difference  in  latitude  between  the  North  and 
South  Poles? 

11.  Two  trains,  one  behind  the  other,  are  moving  in  the  same 
direction  with  speeds  of  45  and  32  miles  an  hour.  What  is  the 
difference  in  their  speeds?  In  answering  the  next  question  does 
it  make  any  difference  which  train  is  ahead?  How  fast  is  the  one 
approaching  the  other? 


NEGATIVE  NUMBERS  83 

12.  The  same  two  trains  are  moving  toward  each  other. 
How  fast  are  they  approaching  each  other? 

13.  An  airplane  is  maintaining  a  speed  of  80  miles  an  hour. 
How  fast  is  it  moving  across  the  country  if  flying  with  a  35-mile 
an  hour  wind?     If  flying  against  a  head  wind  of  35  miles  an  hour? 

14.  If  the  speed  of  the  airplane  is  a  miles  an  hour  and  the  rate 
of  the  wind  is  h  miles  an  hour,  state  the  rate  at  which  the  airplane 
travels  with  and  against  the  wind. 

15.  A  man  in  a  boat  is  rowing  upstream.  He  can  row  at  the 
rate  of  5  miles  an  hour.  What  is  his  rate  upstream  if  there  is  a 
current  of  2  miles  an  hour?  Of  5  miles  an  hour?  Of  6  miles  an 
hour? 

64.  Subtraction — expressions  of  several  terms.     If  the 

number  to  be  subtracted  is  coraposed  of  two  or  more  terms, 
the  same  procedure  is  followed  (see  page  79) :  each  term  is 
subtracted  in  turn.  To  subtract  2:^—4  from  5^^ — 3,  subtract 
each  term  of  2^  —  4  in  turn.  2x  is  positive,  so  add  —2x. 
—  4  is  negative,  so  add  +4  and  obtain 

5^-3-2^+4 

Now  add  terms  of  the  same  kind  and  we  have 

3:^+1 

The  work  may  be  written  in  the  form 

(5^-3)-(2^-4)=5x-3-2:x:+4 
=  3^+1 

Another  illustration  of  this  follows : 

Subtract  —  3:\;+7  from  4:X  —  2 

It  may  be  written 

(4^+2) -(-3x+7) 
—  Zx  is  negative,  so  add  3^ 
+7  is  positive,  so  add  —7 
4^+2+3a:-7 
Add  like  terms,  7x—b 


84  BEGINNERS'  ALGEBRA 

EXERCISES 

1.  From  7a;+3  subtract  2x4-1. 

2.  From  9x+7  subtract  5x+9. 

3.  From  13:r+2  subtract  4x-3. 

4.  From  13:^;— 5  subtract  5^—2. 

5.  From  19^-8  subtract  12:x;+7. 

6.  From  Sx+3  subtract  12x-4. 

7.  From  3a  -5b  subtract  a+76. 

8.  From  5x-2y-\-6  subtract  2x-9y-7. 

9.  From  7n  -3/+ a  subtract  2»+7/  -a. 

10.  (7:i;-3>'+2)-(4x+2>'-5)=  ? 

11.  {ISn-i-lOp  -5)  -i7n-5p  -9)=  ? 

12.  (40^  -605  -7C)  -(50^  -9C+35)  =  ? 

65.  Arrangement  of  work.  It  may  be  desirable,  some- 
times, to  arrange  the  work  of  subtraction  as  in  addition  with 
like  terms  under  each  other. 

From  7x-2y-Q 

subtract  5x-{-3y—2 

2x-5y-4: 

In  subtracting  a  term  you  must  be  very  careful  to  add  a 
term  of  opposite  sign.  The  change  of  sign  is  carried  in  the 
mind  and  not  actually  put  down  on  paper  as  in  the  other 
way  of  doing  the.  work : 

{7x-2y-Q)-{5x+Sy-2)=7x-2y-Q-5x-3y-\-2  = 
2x-5y-4 

66.  Double  use  of  word  negative.  We  call  2x  sl  positive 
term  and  —  2;c  a  negative  term.  The  two  terms  have  oppo- 
site signs.  We  often  say  that  —2x  is  the  negative  of  2x, 
We  may  also  say  that  2x  is  the  negative  of  the  term  —2x. 
That  is,  we  may  use  the  word  negative  to  mean  opposite  in 
sign.     This  is  a  very  reasonable  use  of  the  word,  for  we  do 


NEGATIVE  NUMBERS  85 

not  know  whether  2x  stands  for  a  positive  number  or  a 
negative  number,  for  that  depends  upon  whether  x  is  3. 
positive  or  negative  number.  So  a—h  and  —  a+&,  or 
b  —  a,  are  to  be  regarded  as  the  negatives  of  each  other. 
4—2%  is*the  negative  of  2x—4:.  The  negative  of  —Zx—2 
is  3a;+2.  The  rule  for  subtraction  may  then  be  put  in  the 
more  concise  form : 

Rule.    To  subtract  a  number  add  its  negative. 

EXERCISE    I 

1.  Subtract  lx-2y+^xy-b  from  Zbx-{-lly -l2xy-\-b. 

2.  From  ^x-2xy-\-b-by  subtract  7x-2y-^Zxy-^. 

3.  Subtract  ny-\-^x-\-b-2b-{-a  from  19y-6x+2a-7fe-6. 

4.  From  3a&-26c+6ca-9  subtract  5a6-2ca4-76c+6. 

5.  Subtract  7x-2la-Sb-2c  from  9x-10a-{-7b-\-c. 

6.  From  3aa-7a+3  subtract  2aa-2a-h7. 

7.  From  2a  -5b-\-d  subtract  7a -9c-  15d-{-f. 

8.  Subtract  2U  -S5xy+9  from  4:x-{-2xy -13, 

9.  Subtract  ia+^ft  from  ^a  -^b. 

10.  Subtract  Sx-2y-.5z  from  2x-y-.2z. 

11.  From  Qx-7y-3z-\-4:a  subtract  9x+Qy  -4:z-\-5a. 

12.  From  Sx-Sy-\-9z-5a  subtract  6jc -4^+102; -3a. 

13.  From  Sx—  .5y+.7z  subtract  4:X-\-8y  —  .5z. 

Add  the  following  expressions  and  from  the  sum  subtract  each 
expression  in  turn  imtil  nothing  is  left: 

14.  3jc-4y+2z  15.  2x-  y-  z 
-8ic+5>'-72  a;+2>'-2z 
+2x  -Qy+Sz  -3x  -3>'+3z 
—5x-{-5y—Qz  a:— 43'+5z 
— ac— 3y-f4z  —2x-^5y—^ 

In  an  exercise  when  there  are  many  niunbers,  some  to  be 
added  and  some  to  be  subtracted,  all  the  subtractions  may  be 


86  BEGINNERS'  ALGEBRA 

replaced  by  additions  and  the  exercise  treated  as  suggested 
in  the  article  on  addition. 

Add  Sx,  subtract  2x,  subtract  —3,  subtract  —7x,  add  5x, 
subtract  x,  subtract  —4,  write  thus: 

dx-2x+S+7x-{-5x-x-{-4:  =  12x+7 
So  also,  to  V^x—2  add  4^—3,  subtract  5^+2,  subtract 
2a;  — 7,  write  thus: 

13:x;-2+4^-3-5x-2-2%+7=  ? 

16.  Add  7jc— 3,  subtract  2x—^,  subtract  bx—2. 

17.  Add  13»+5,  subtract  9«-6,  subtract  7w+7. 

18.  Subtract  ^x  -2,  subtract  14x  -3,  add  9:c+7,  subtract  lOx-\-^. 

19.  Add  x-h-\-a,  subtract  2x-Zb-\-2a,  subtract  -3x+2&-5a. 

EXERCISE   II 

Solve  and  check: 

1.  3;r-5  =  7a;-10 

2.  9a:+8  =  4:«+23 

3.  5:r+17  =  10:«-3 

4.  Sx-\-l-^x  =  ^x-l^-\-Zx 

5.  3:c+8-5^-2  =  6x+4-ic-19 

6.  3y-12+y+6  =  24-83'-2+5y 

7.  Gjc-9+;c+4  =  3x+15-4x 

8.  5a+10-7o+2  =  25-4a-3  i 

9.  7a-4-2a-3  =  8a+4-a-19 

10.  x+ 16 -5:i; -2  =  18jc -4-10:^-6 

11.  4-»+7-3w=15-8w+4+2w 

12.  :r-8+2jc-2  =  6-3x-2-8jc 

13.  2;c-3-(.'»:+5)  =  3a;-12 

14.  4y-9  =  12-(y-3)  +  l 

15.  14-(:«-2)  =  27-(2+4:»;) 


NEGATIVE  NUMBERS  87 

16.  3w-(»-5)  =  16-(3+2w) 

17.  27 -(x -3)  =  15 -(5 -4a;) 

18.  4/>-3-(9-3/>)  =  9 

19.  3a+9  =  15-(a-8)+2 

20.  5jc+6-20jc  =  3-5x-(5a;-2) 

21.  6:«:-(4x-3)  =  8^ 

22.  10x-{x-2)  =  2-{Qx-5) 

23.  3x-{x-a)  =  b 

24.  Using  the  answer  to  Exercise  23,  find  answers  for  the  special 
equations  when: 

(1)  a  =  2,      b  =  Q 

(2)  a=-5,  6=11 

(3)  a  =  3,       b=-S 

25.  5x=b-ia-2x) 

26.  ^x-ix-b)  =  2b-{x-^a) 

PROBLEMS 

1.  I  have  three  numbers;  the  second  is  2  more  than  the  first, 
and  the  third  is  4  less  than  twice  the  first.  If  their  sum  is  34,  find 
the  numbers. 

2.  Of  the  three  sides  of  a  triangle,  the  second  is  3  inches  more 
than  the  first,  and  the  third  2  inches  less  than  the  first.  If  the 
perimeter  is  22  inches,  find  the  sides.  » 

3.  Of  the  angles  of  a  triangle,  the  first  is  20°  less  than  the  second, 
and  the  third  is  40°  less  than  twice  the  second.  Find  the  number 
of  degrees  in  each. 

4.  The  length  of  a  certain  rectangle  is  6  feet  less  than  4  times  its 
width.     If  the  perimeter  is  108  feet,  find  the  dimensions. 

5.  If  4  more  than  a  certain  number  is  subtracted  from  4  times 
the  number,  the  result  is  26.     Find  the  number. 

6.  If  3  times  a  certain  number  is  subtracted  from  the  sum  of  the 
nimiber  and  20,  the  result  is  4.    Find  the  number. 

7.  If  3  is  subtracted  from  a  certain  number  and  this  remainder 
is  subtracted  from  4  times  the  number,  the  result  is  18.  Find  the 
number. 


^  BEGINNERS'  ALGEBRA 

8.  If  2  is  added  to  4  times  a  number  and  the  sum  is  subtracted 
from  6  times  the  number,  the  result  is  20.    Find  the  number. 

9.  If  3  is  subtracted  from  7  times  a  number,  and  the  difference 
is  subtracted  from  10  times  the  number,  the  result  is  33.  Find 
the  nimiber. 

10.  Of  the  three  angles  of  a  triangle,  the  second  is  20°  less  than  the 
first,  and  the  third  is  12°  more  than  the  sum  of  the  other  two. 
Find  the  number  of  degrees  in  each  angle. 

11.  There  are  two  consecutive  integers  such  that  if  the  larger 
is  subtracted  from  twice  the  smaller  the  result  will  be  14.  Find 
the  numbers. 

12.  There  are  two  consecutive  integers  such  that  if  4  times  the 
smaller  is  subtracted  from  the  larger  the  result  is  —26.  Find  the 
numbers. 

13.  There  are  three  consecutive  integers  such  that  if  the  third  is 
subtracted  from  the  sum  of  the  first  two  the  result  is  9.  Find  the 
numbers. 

14.  There  are  three  consecutive  integers  such  that  if  the  second 
is  subtracted  from  the  sum  of  twice  the  first  and  the  third  the  result 
is  31.    Find  the  numbers. 

15.  There  are  three  consecutive  integers  such  that  if  3  times  the 
first  is  subtracted  from  the  sum  of  the  second  and  the  third  the 
result  is— 16.    Find  the  numbers. 

16.  The  length  of  a  certain  rectangle  is  2  feet  more  than  its 
width.  The  side  of  a  certain  square  is  twice  the  width  of  the  rec- 
tangle. The  perimeter  of  the  square  is  16  feet  more  than  the 
perimeter  of  the  rectangle.    Find  the  dimensions  of  each. 

67.  Multiplication.  We  know  that  in  arithmetic  the 
product  of  5  times  3  is  the  same  as  the  product  of  3  times  5; 
that  is, 

5X3=3X5 

The  order  in  which  the  multipl3Kng  is  done  makes  no 
difference  in  the  result.  The  idea  is  just  as  true  in  algebra. 
Whatever  may  be  the  product  of  3  times  —5,  —5  times  3 
must  give  the  same  result.  With  this  understanding  three 
cases  arise  in  multiplying  positive  and  negative  numbers: 


NEGATIVE   NUMBERS  89 

A  positive  number  times  a  positive  number,  (+3)  X  (+5) 
A  positive  number  times  a  negative  number,  (+3)X(  — 5) 
A  negative  number  times  a  negative  number,  (— 3)  X(— 5) 
How  to  handle  the  first  case  is  known  to  us  in  arithmetic : 
(+3)X(+5)  =  +15 

We  may  think  of  it  as  adding  5  three  times.  On  the  num- 
ber scale  this  would  mean  moving  to  the  right  from  zero 
5  unit  places  3  times,  and  give  +15.  The  product  of  a 
positive  3  and  a  positive  5  is  a  positive  15. 

In  like  manner,  on  the  number  scale  (3)X(  — 5)  would 
mean  adding  —5  three  times,  or,  what  is  the  same  thing, 
subtracting  +5  three  times,  that  is,  moving  to  the  left 
5  places  three  times,  and  give  —15.  Thus  the  product  of 
positive  3  and  negative  5  is  negative  15. 

So  also  (— 3)X(— 5)  might  be  thought  of  as  subtracting 
—5  three  times  or,  what  is  the  same  thing,  adding  +5  three 
times,  which  would  give  +15.  Thus  the  product  of  negative 
3  and  negative  5  would  be  positive  15. 

These  illustrations  suggest  the  following  definitions :  The 
product  of  two  positive  numbers  is  a  positive  number. 
The  product  of  two  negative  numbers  is  a  positive  number. 
The  product  of  a  positive  number  and  a  negative  number  is  a 
negative  number. 

Thus,  to  multiply  two  numbers  together,  find  the  product 
of  their  absolute  values  and  write  in  front  of  this  product 
the  proper  sign. 

Insymbols(+3)X(+5)  =  +15,     (+3)X(-5)  = -15 
(-3)X(-5)  =  +15,     (-3)X(+5)  =  -15 

EXERCISES 

1.  (3)X(+7)=  ?      2.  (-3)X(7)=  ?      3.  (-5)x(-ll)=  ? 
4.  9x(-3)=  ?         5.  (-8)X(-9)=  ?  6.  (-7)  •  (a)=  ? 
7.  (-a) -(-5)=?   8.  (-a)'{-b)=  ?  9.  a- (-6)=  ? 
10.  (-a)  •  (-6)=  ?  11.  »  •  (-7)-  ?       12.  -X  •  (-y)=  ? 
7 


90 


BEGINNERS*  ALGEBRA 


13.  Copy  and  fill  in  the  blanks  in  the  following  table,  placing  the 
products  of  a  number  from  the  horizontal  row  with  a  number  from 
the  vertical  column  at  the  intersection  of  its  row  and  column. 


0 

2 

-5 

f 

-6 

2i 

n 

3 

-2 

i 

.7 

-8 

—a 

14.  (-3)(-6)(6)=? 

15.  (-10)(i)(-f)(-f)=? 

16.  (-4)(-3)(-a)=  ? 

17.  (-5a)(3)(-26)=  ? 

18.  (-6)(-3)(+a)(-&)=  ?    ■ 

19.  (-7)(-a)(-a)(-2)=? 

20.  (-5) (-9)  (+3)=  ? 

21.  (_2)(-3a)(-46)(-7)=  ? 

22.  {+x){^9){-Sx){-7)=  ? 

23.  (+5a)(-3:x;)(-7)=  ? 

24.  5( -:*;)( -2)(a)=  ? 

25.  7(-»)(+3/)(-2)=  ? 

68.  Multiplying  a  sum.  In  arithmetic  we  multiply  7+3 
by  5  in  two  ways.  We  may  add  the  7  and  3,  getting  10,  and 
then  multiply  by  5,  getting  50,  or  we  may  multiply  7  by  5, 
getting  35,  and  multiply  3  by  5,  getting  15,  and  then  add 
35  and  15,  getting  50.     If  we  wish  to  express  the  example 


NEGATIVE  NUMBERS 


91 


in  symbols,  we  must  inclose  the  7+3   in   a  parenthesis, 
thus: 

5(7+3)=5- 10  =  50 
5(7+3)  =35+15  =  50 

The  second  way  is  exactly  what  you  do  when  you  multiply 
32  by  3: 

32  is  30+2 
3(30+2)  =90+6 
=  96 

though  you  usually  write  it 

32 

96 

Rule.    To  multiply  a  sum  by  any  nimiber,  multiply  each 
term  of  the  sum  by  the  number. 

EXERCISES 


Work  both  ways  when  possible: 


1.  5(7+4) 
3.  3(8+2-5) 
5.  7r(9+4) 
7.  (-2) (4+3) 
9.  (-3)  (7 -2+3) 
11.  3(a-4) 
13.  5{x+y-7) 
15.  3x(a-7) 
17.   -4.{x-y) 
19.  5aia-x-y) 
21.  7yia-2b+3y) 
23.   -2(3 -5a;) +2 


2.  5(7-4) 

4.  6(7r-3)     (Use7r  =  3.142) 

6.  7r(9-4) 

8.  (-2) (6 -3) 
10.  5(8-10+2) 
12.  -a(:r-4) 
14.  20(x-y\-3) 
16.  Tr{a-b) 
18.  4xix-a+2) 
20.   -7x(a'2x+^) 
22.   -12x(a-46+3:c) 
24.   -l(-7+x-36) 


92  BEGINNERS'  ALGEBRA 

It  is  to  be  noticed  that  when  we  multiply  a  sum  by  a 
number  we  introduce  the  number  as  a  factor  in  each  term 
of  the  sum : 

2(3+7)  =2X3+2X7 
2ia-b)=2a-2b 

But  when  we  multiply  a  product  by  a  number  we  simply 
introduce  it  as  a  factor  but  once : 

2a(xy)  =2axy 
2(3-7)=2-3-7 

It  should  also  be  noticed  that  the  result  is  the  same  in 
whatever  order  we  combine  the  factors : 

2X3X7  =  6X7  =  42 

3X7X2  =  21X2  =  42 

7X2X3  =  14X3  =  42 
Find  the  values  when  a  =  2,  b=  —2,  c  =  3,  x=  —4,  y=^  —5,  z  =  6: 
25.  a{b-c)  26.  b{a-c-x)  27.  b{c-x+y) 

28.  .^(^+53; -42)      29.  x(ab-2ay+bc)        30.  bx{a+Sy-2) 
31.  ax{b-5y-ic)     32.  bc{Sai-2y -2b)        33.  a{bcx) 
34.  axibayz)  35.  b{a-{-b+c)  36.  {abc){xi-y) 

69.  Division.  Multiplication  is  the  operation  of  finding 
the  product  when  the  factors  are  given.  Division  is  the 
operation  of  finding  one  of  the  factors  when  the  product 
and  the  other  factor  are  given. 

7X3=? 
7X?  =  35 

Because  of  this  relation  division  is  called  the  inverse  of 
multiplication.  Show  that  this  fact  enables  us  to  write 
down  the  following  rule  for  dividing  positive  and  negative 
numbers : 

Rule.    The  quotient  of  two  positive  numbers  is  a  posi- 
tive number. 
The  quotient  of  two  negative  numbers  is  a  positive  nimiber. 


NEGATIVE  NUMBERS 


93 


The  quotient  of  a  positive  number  and  a  negative  number 
is  a  negative  number. 


(+7)X(     )  =  +28 

(     )x(9)=45 

(+7)X(     )  =  -28 

(     )x(9)  =  -45 

(-7)X(     )  =  +28 

(   )x(- 

-9)  =  +45 

(-7)X(     )  =  -28 

(   )x(- 

EXERGISES 

-9)  =-45 

Divide: 

1.  45  by  5                 2. 

72  by  -12 

3.   -96  by  3 

4.   -25  by  -50        5. 

91  by  -91 

6.  t  by  -i 

7.   -Aby-i^        8. 

-2.3  by  7 

9.  3.92  by -1.33 

10.  a  by  -a              11. 

—a  by  —a 

12.  2a  by  2 

13.  2a  by  -2           14. 

3a  by  —a 

15.  6oby  -2a 

70.  Mastery  of  the  laws  of  signs. 

A  thorough  mastery 

of  the  laws  of  signs  for  the  four  fundamental  operations  is  of 
great  importance.  There  should  be  no  hesitation  in  your 
mind  as  to  the  proper  sign  to  be  used  in  any  case  that  comes 
up.  Practice  these  operations  luitil  they  become  automatic. 
Such  mastery  will  save  you  much  trouble  in  the  futiu-e. 

EXERCISES 

1.  Copy  and  fill  in  the  blanks  of  this  table  using  the  values  of  a 
and  h  given  in  the  first  two  columns. 


a 

h 

a  +  h 

a-  b 

ab 

a 

~b 

12 

4 

9 

15 

8 

-4 

-9 

+3 

-7 

-4 

94  BEGINNERS*  ALGEBRA 

Evaluate,  that  is,  find  the  value  of,  the  following  if  a=  15,  6  =  8, 
c=2,  x  =  3y  y  =  5,  working  the  first  four  exercises  in  more  than  one 
way: 


-1 

3.  "'" 

y 

4.  ^           5.  t^            6.  I 

'■  H 

<-\ 

'■^     'o-l+H 

11.  Work  the  same  exercises  when  a  =15,  6=  —8,  c  =  2,  x=  —3, 
y  =  5. 

Divide: 

12.  127rby4  13.  IbirahySa 
14.  SSx  by  -3  15.   -2^x  by  8 

16.   -18ajcby9a  17.  +15bxby  -3b 

71.  Dividing  a  sum.     Inarithmetic  wemay  divide27— 15 
by  3  in  two  ways. 

Thus,  (27-15)^3  =  12-^3  =  4 

or  (27-15)-^3  =  (27-^3)-(15■f-3) 

=9-5=4 
A  better  manner  of  expressing  it  is 
27-15_12^ 
3  3 

27-15    27     15     ^     -     ^ 
-3-  =  ^--  =  9-5  =  4 

Rule.    To  divide  a  sum  by  a  number,  divide  each  term 
of  the  sum  by  the  number. 

EXERCISE    I 

Divide : 

1.  (81  -27)  by  9  2.  97r  -Gtt  by  3 

3.  97r-67r  by  37r  4.  27a -15a  by  3 

5.  45:x:-273C  by  9  6.  9ic+15  by  3 

7.  4jc-12by  -4  8.   -3:«+21y  by  -3 

9.  3a.r-2ay+5a  by  a  10.  3Qax-9x  by  9x 


NEGATIVE  NUMBERS  95 

11.   -3Qax-\-9x  by  -Sx  12.  3:r+8  by  3 

13.  5a;-7by2  14.  7.^-8  by  -3 

15.   -2a-3&+5(;by-3 

EXERCISE    II 

Evaluate  when  x  =  4:,  y=  —3,  z=S,  o  =  2,  b=  —3: 

x+2y+3z  3x-12y+62;  tfx-4fl 

a  6  2 

ax—4:by-{-z  ay—ix-\-z  2a-\-x-\-2y 

4-  i: 5.  7 —  0.  7 — 

ah  a  —  b  ab 

If  fl  =  2,  b=  —3,  c  =  5,  a;=  —4,  find  the  values  of  the  expressions: 

7.  2{a-b)  8.  5(a-c)  9.  fl(6-c) 

10.  ^  11.  ^x  12.  ^^ 

io    <^+&  ,  «-c                ..    2a—Sb-\-2c                    ab-\-bcJ-ca 
1^-  "2~+"T"  ^^-      4a -26  1^-  ^ 

72.  Equations.     It  may  save  you  some  trouble  in  chang- 
ing the  forms  of  such  expressions  as 

(1)  x+2{2x-5) 

(2)  x-2{5x-2) 

if  you  take  care  to  read  out  in  words  the  meaning  of  the 
expressions.  (1)  means  that  2  times  each  term  in  the  paren- 
t.hesis  is  to  be  added  to  x,  while  (2)  means  that  2  times  each 
term  in  the  parenthesis  is  to  be  subtracted  from  x.  Remem- 
ber also  what  is  to  be  done  when  one  number  is  to  be 
subtracted  from  another  in  algebra. 

EXERCISES 

Solve  and  check: 

1.  :c-|-9  =  2  2.  a;4-ll  =  4 

3.  «+13  =  7  4.  3x+14  =  2xH-10 

5.  2^+10  =  ic+8  6.  jc+15=10 

7.  :»+9  =  2ic-i-15  8.  «-h5  =  2ic+10 


96  BEGINNERS'  ALGEBRA 

9.  4*+15  =  3x+12  10.  5x+lQ=4x+lS 

11.  x-a=b 

12.  Derive  equations  from  Exercise  11  in  which: 

(1)  o=12,      6=6  (2)a=-9,  6=2 

(3)  a=-13,  6=-3        (4)  a  =10,    6  =  8 

13.  Find  the  answers  to  the  equations  obtained  in  Exercise  12 
by  adding  the  numbers  as  indicated  in  the  answer  to  Exercise 
11. 

14.  If  a  is  subtracted  from  a  certain  number  and  this  remainder 
is  subtracted  from  twice  the  number,  the  result  is  4(i.  Find  the 
number. 

15.  Solve  Exercise  14  when  a  is  5,  20,  -6,  -3,  9,  -18,  15. 

16.  6jc+4(:*;-2)  =  22  17.  8ic+2(:r-5)=90 
18.  6jc-3(x-4)  =  36  19.  157t-2(3w-8)  =  34 
20.  8:»;-2(2:i;-5)=2  21.  3«-4(2»-9)  =  6 
22.  7«+2(3w+14)  =  2  23.  33;-5(3y+2)  =  38 
24.  5x-2(5:x;+20)  =  10  25.  7/+3(5/+l)=  -8 

26.  9:»-3(:*;-12)=-6  27.  4(3»-5) -5(5w-7)=2 

28.  12:*;  =  56 -4(^-6)  29.  9p  =  12+Z{p-S) 

30.  2ic-7(25~ic)  =  2(25-3^) 

31.  S{x-S)-iQ-2x)=2{x-\-2)-5(5-x) 

32.  2(3;-3)-5(>'-l)  =  7+(2-3') 

73.  The  fraction  as  a  parenthesis.  In  the  solving  of  the 
equation 

x-2_x 
5    ~6 
the  numerator  x  —  2  must  be  regarded  as  one  number.     Thus 
when  we  multiply  both  sides  of  the  equation  by  30  we  write 

6(^-2)  =5;c 

The  fractional  sign  acts  the  same  as  a  parenthesis : 

6:x;-12  =  5:x: 


NEGATIVE  NUMBERS      •  97 


In  the  same  way  we  have 


3(^-2)_ 
"^  5      "^ 

5A;-3(it;-2)=20 

2^=14 

x  =  7 
x-2     . 


and  also  ^~    'i 

3:^- (^-2)  =  12 
3^-::c-f2  =  12 
2:x;  =  10 
x  =  5 

EXERCISES 

Solve  and  check: 
,    X4-32     X  o    3a     .._2a 

3.1^:3^=1  4.4-^=1 

4  4 

_  a;  — 1     -.  c    ^     w— 3     p. 

5.  0=-- 3-  =  5  6.  2— 5-  =  9 

7.  «_^±?  =  2  8.7-5+15=5 

3       6  5 

9.  Il-^±i5=_2^  10.  9-^  =  12 

2  o 

5  0       0 

13.  *-Eri4=4  14.  £±3_£-3^g 

5        4  2  5 

15.  ^-^-^  =  1  16.  4«-4?  =  l 

4  3  8  0 

17    ^^-^-i    3»+17  jg   ^JlZ=io    4(^-1) 


4  10  6  9 

-2    c+2     c-l_^  ^^    a:-4    x-S_x- 

_____  __o  JO.  ^  -  ^ 


98  BEGINNERS'  ALGEBRA 

74.  Problems  with  impossible  answers.     A  problem  to  be 
solved  is  stated  in  the  form  of  an  equation.     The  equation 
is  solved;  that  is,  a  satisfactory  value  for  the  unknown  is 
found.     It  does  not  always  follow  that  the  answer  of  the 
equation  will  serve  as  an  answer  to  the  problem.     For 
instance,  consider  the  problem:    The  sum  of  two  consecutive 
integers  is  10,  what  are  the  integers? 
:x:+(a;+1)  =  10 
2x  =  9 
9 

9 

-  satisfies  the  equation,  but  is  not  a  correct  answer  to  the 

9  .  . 

problem,  for  ^  is  not  an  integer.  The  answer  to  this  equa- 
tion when  used  as  an  answer  to  the  problem  is  nonsensical. 
If  there  were  integers  of  the  kind  sought  in  the  problem,  the 
solution  of  the  equation  would  have  given  them.  The  solu- 
tion of  the  equation  shows  there  are  no  such  integers.  The 
problem  is  an  impossible  problem;  that  is,  it  has  no  solution. 

This  illustration  shows  the  great  importance  of  trying  the 
answer  obtained  from  the  equation  in  the  problem  itself  to 
see  if  it  actually  works.  It  is  often  quite  as  important 
to  learn  that  a  given  problem  cannot  be  solved  as  to  be  able 
to  find  the  answer  to  one  that  can  be  solved.  The  roots  of 
equations  arising  from  problems  should  always  be  inter- 
preted by  being  applied  to  the  problem  itself. 

To  illustrate  this  in  another  way:  Two  sides  of  a  certain 
triangle  were  found  to  be  respectively  1  and  2  inches  longer 
than  the  third  side.  The  sum  of  the  two  shorter  sides  is 
3  inches  shorter  than  the  longest  side!  What  are  the  sides 
of  the  triangle? 

The  equation  is 

shortest  side + next  larger  =  largest — 3 
x+ix+l)  =  {x+2)-S 
whence  x=—2 


NEGATIVE  NUMBERS  99 

But  —2  has  no  meaning  when  used  as  the  side  of  the  tri- 
angle. The  problem  is  impossible.  Clearly  the  man  who 
made  the  measurement  made  some  mistake  in  furnishing 
the  data  for  the  problem. 

PROBLEMS 

Interpret  carefully  the  results  obtained  in  solving  the  following 
problems  and  determine  which  problems  really  have  answers: 

1.  Is  there  a  number  such  that  the  result  of  subtracting  3  times 
itself  from  itself  will  be  15  more  than  itself? 

2.  Three  times  an  integer  minus  5  times  the  next  larger  integer 
is  37.    Find  the  nimiber. 

3.  Find  an  integer  such  that  5  times  it  plus  2  is  49. 

4.  There  are  three  consecutive  integers;  if  5  times  the  middle  one 
is  subtracted  from  3  times  the  sum  of  the  other  two,  the  result 
will  be  4.    Find  the  nimibers. 

5.  There  are  four  consecutive  integers;  if  twice  the  sum  of  the 
first  and  third  is  subtracted  from  3  times  the  sum  of  the  second 
and  fourth,  the  result  will  be  24.     Find  the  numbers. 

6.  There  are  three  consecutive  integers;  if  the  first  is  subtracted 
from  twice  the  third,  the  result  is  3  times  the  second.  Find  the 
numbers. 

7.  The  sum  of  three  consecutive  integers  equals  the  middle  one. 
What  are  the  nimibers? 

8.  1  have  $4  in  dimes  and  quarters.  If  I  have  25  pieces  of  money 
altogether,  how  many  of  each  have  I 

9.  I  have  $5  in  dimes  and  quarters,  having  30  pieces  in  all.  How 
many  of  each  have  I? 

10.  Twenty-five  dollars  were  to  be  divided  between  two  boys 
in  the  ratio  of  1  to  3.    What  was  the  share  of  each? 

11.  Twenty-five  electric  light  bidbs  were  to  be  divided  between 
two  boys  in  the  ratio  of  1  to  3.    What  was  the  share  of  each? 

12.  Two  boys  were  talking  about  their  ages;  the  older  boy  said, 
"  I  am  3  times  as  old  as  you,  but  in  5  years  I  shall  be  only  twice  as 
old  as  you.     How  old  am  I?"  , 


100  BEGINNERS'  ALGEBRA 

13.  The  younger,  not  to  be  outdone,  said,  "All  right,  but  I  have 
a  brother  who  is  3  times  as  old  as  I,  and  in  5  years  he  will  be  5  times 
as  old  as  I.    How  old  is  he?" 

14.  One  side  of  a  triangle  is  2  inches  longer  than  a  second  side, 
the  second  side  is  4  inches  longer  than  the  third  side,  the  perimeter 
is  7  inches.    What  are  the  sides  of  the  triangle? 

15.  The  sides  of  a  triangle  have  the  same  relation  to  one  another 
as  in  the  last  problem,  but  the  perimeter  is  22.  What  are  the  sides 
of  the  triangle? 

16.  Find  the  dimensions  of  a  square  such  that  when  one  side  is 
increased  by  21  inches  and  the  adjacent  side  decreased  by  3  inches 
the  perimeter  of  the  resulting  rectangle  is  34. 

17.  A  certain  mixture  of  two  substances  containing  5  times  as 
many  poimds  of  one  substance  as  of  the  other  weighs  45  pounds. 
What  is  the  quantity  of  each  substance? 

18.  In  a  certain  lot  of  eggs  there  were  twice  as  many  bad  eggs  as 
good  ones.  There  were  40  in  all.  How  many  good  eggs  were 
there? 

19.  There  are  three  consecutive  even  numbers;  the  sum  of  the 
first  two  is  3  times  the  third.    What  are  the  numbers? 

20.  There  are  two  consecutive  even  numbers  whose  simi  is  3 
times  the  number  midway  between  them.    What  are  the  numbers? 

21.  In  a  lot  of  7  dozen  eggs  the  ratio  of  the  good  to  the  bad  was 
2  to  3.    How  many  good  eggs  were  there  in  the  lot? 


CHAPTER  V 


Graphics 


75.  Comparison  of  numbers.  In  our  everyday  life  we  are 
frequently  called  upon  to  compare  numbers,  such  as  the 
population  of  cities,  the  size  of  crops,  records  in  athletics, 
prices  of  goods.  It  has  become  a  very  common  practice  to 
represent  numbers  by  lines  or  bars  drawn  to  some  convenient 
scale.  The  eye  can  more  readily  compare  a  lot  of  lines  than 
the  numbers  in  a  table. 

Compare  the  two  ways  of  presenting  the  number  of  acres 
planted  in  the  following  crops  in  the  United  States  for  the 
year  1919: 


Acres 

Com. . . 

..102,075,000 

Oats . . . 

..  42,400,000 

Wheat. . 

..  73,243,000 

Barley. . 

. .     7,420,000 

Hay.... 

..  72,034,000 

Cotton . 

..  33,344,000 

lukeat 

barley 

hay 


millions  of  aertz 
10     20     30     40     50     60 


70     80     90    lOO 


Fig.  11 

To  present  a  table  of  numbers  graphically  in  this  way, 
some  convenient  unit  is  chosen,  and  lines  are  drawn  to  scale. 
The  lines  or  bars  may  be  drawn  in  either  a  vertical  or  a 
horizontal  position.  It  will  help  to  make  the  comparison 
clearer  if  the  bars  are  arranged  in  the  order  of  length  if  there 
is  no  other  order  of  arrangement  suggested.  The  scale 
should  be  marked  on  paper  as  in  the  figure  given.  Where 
the  numbers  are  large,  approximate  values  should  be  used. 
Round  the  numbers  off  to  convenient  size  for  plotting.  In 
the  illustration  the  numbers  are  used  to  the  nearest  million 
acres. 


101 


'  '  lOi  '     '  BEGINNERS'  ALGEBRA 


EXERCISES 

1.  Compare  the  lengths  of  the  following  important  rivers  by- 
means  of  lines.     Use  ^3^  or  |-  of  an  inch  to  the  100  miles. 

Miles  Miles 

Mississippi 2800  Amazon 3600 

Rio  Grande 1800  Danube 1800 

Yukon 1600  Nile 3900 

Columbia 1200  Congo 1600 

2.  Show  by  a  bar  diagram  the  comparative  heights  of  the  follow- 
ing structures: 

Feet  Feet 

Eiffel  Tower 984  Woolworth  Building  729 

Washington  Monu-  St.  Peter's,  Rome. .  435 

ment 555  Pyramid  of  Cheops.  451 

3.  Use  a  bar  diagram  in  comparing  the  number  of  telephones 
per  hundred  people  in  the  following  countries: 

Argentina 1.1  Great  Britain 1.9 

AustraHa 4.0  Italy 3 

Canada 8.1  Norway 4.5 

Denmark 7.3  Sweden. 6.4 

France 9  Switzerland 3.0 

Germany 2.3  United  States 11.4 

76.  Use  of  cross-ruled  paper.  The  making  of  such  dia- 
grams is  much  easier  when  cross-ruled  paper  is  used.  A 
scale  is  chosen  and  marked  along  one  edge.  The  length  of 
the  bar  is  estimated  as  nearly  as  possible. 

The  following  table  and  diagram  show  the  density  of 
population  of  a  number  of  states — that  is,  the  average 
number  of  people  to  the  square  mile.  Let  the  student 
complete  the  figure. 

Massachusetts 479  Illinois 116 

Rhode  Island 566  Michigan 64 

Connecticut • . .  286  Kansas 22 

New  York 218  Texas 18 

Pennsylvania 195  Colorado 9 

Ohio 141 


GRAPHICS 


103 


!  j        1  ■      '          M                

I"     ."T- 

_     :       : : :  _     -i-i. 

1  i  t    1                                    ..---- 

_  '    ._      ._      _   _ 

" "    ""                      '"  ,  i  1  ;  1  1  II  1  1  i  1  1  M  1  1  1  M  1  1  1  1  1  1  M  1  1  1  1  1 

600  ■: 

I  I                                  i--    ■ 

i      ' 

i                                   ■  t             

1 

--    -L    --      ;  1  1 

\  •       !  1         ■     ---     -p-4-  ■-        -     -        - 

1  !     I!  i  i  '  '  M  '     i  1        -p                   --    - 

i  j 

;  I       !  ■       t              .--.i- --  —  -        _. 

!    '  ' 

Me\f\                            < 

!l  1  i  '  1  i        'II                        1                     -.-.-.. 

■*" 

Ml,         '  ■  1         ....           -      .-..      - 

"  " " 

1  !    '  i  '  ,    ;  1  I         ' 

1  ,  i  1    1  '      i  i  ■  1 1      '      ' 

1  M   1  M     i     !  1 1 '     ' '  ■  r          

1.             '1 

1     1      1           ;      1      1     1           '      1            •     •                                  :      .      ■               ,           ■      '                     '     ! 

!           I           '      i      M                '':'•■           \                       i      ^      •          ,     1           i     1           i           !           ' 

\(\(\ 

100             H:-; 

i  1      ,       — T  !  :  ■  i  '       r    '  '  i    !  1    I  M  M 

'                                                  ,                          1     :          ■     1     1          1     !          1     i     1     M 

...    ^  T  ^ 

1 

1  ,    .                     1            T        T         1 

hi 

hi 

1  1  1  1 

i                   1  1 

1  1     l-l.I  111              1            11 

P^ 


%    o    ^ 


Fig.  12 


EXERCISES 


1.  Display  the  comparative  heights  of  mountains  listed  below. 
Use  scale  1  inch  to  4000  feet.  Taking  numbers  from  the  table  to  the 
nearest  hundred  feet,  arrange  in  order. 


Feet 

St.  EHas 18024 

Ranier 14408 

Pike's  Peak 14109 

Popocatepetl 17844 

Chimborazo 20517 


Feet 

Blanc 15780 

Vesuvius 4267 

Fujiyama 12395 

Kilauea 4040 

Everest 29141 


2.  Compare  on  cross-ruled  paper  the  population  of  the  ten 
largest  cities  in  the  United  States,  census  of  1920.  Use  numbers 
to  the  nearest  10,000. 


New  York  City . .  5,621,151 

Chicago 2,701,705 

Philadelphia....  1,823,158 

Detroit 993,739 

Cleveland 796,841 


St.  Louis 772,897 

Boston 748,060 

Baltimore 733,826 

Pittsburgh 588,193 

Los  Angeles 568,886 


104 


BEGINNERS'  ALGEBRA 


77.  Diagrams  involving  time.  In  the  exercises  we  have 
been  considering  there  was  no  particular  order  other  than 
magnitude  in  which  the  bars  should  be  arranged.  With 
statistics  taken  at  stated  intervals  of  time  there  is,  however, 
a  natural  order.  The  bars  should  be  arranged  in  the  order 
of  tin:^e  and  at  proper  intervals.  Two  scales  should  be 
marked  off,  preferably  at  the  left  and  lower  edges  of  the 
diagram,  as  in  Fig.  13,  which  shows  the  percentags  of  the 
total  population  of  the  United  States  living  in  cities  of  over 
8000  inhabitants. 


Year 

1790. 
1800. 
1810. 
1820. 
1830. 
1840. 
1850. 
1860. 
1870. 
1880. 
1890. 
1900. 
1910. 
1920. 


Per  Cent 

.  3.4 
.  4.0 
.  4.9 
.  4.9 
.  6.7 
.  8.5 
.12.5 
.16.1 
.20.9 
.22.6 
.29.2 
.33.1 
.39.0 
.43.8 


30 


20 


""   t"'H 

S 

::: 

._. 

-  -  .-- 

:: 

W 

;:: 

::: 

;; 

1 

::: 

:: 

:\ 

3;: 

|: 

1 

4--^ 

::::  +  : 

$ 

rl 

U- 

r 

H|i 

;:  : 

;;; 

: 

r-,  r::. 

— 

„ 

_  —  - 

_ 

__ 

rs 

n  rs 

oooooooooooovovX>vo  iiaart 
ooooooooo' 

Fig.  13 


EXERCISES 

1.  Display  by  means  of  a  bar  diagram  the  growth  of  population 
of  the  United  States.  Use  the  numbers  to  the  nearest  hundred 
thousand.     Draw  the  bars  vertically. 


1790 3,929,214 

1800 5,308,483 

1810 7,239,881 

1820 9,033,453 

1830 12,866,020 

1840 17,069,453 

1850 23,191,876 


1860 31,443,321 

1870 38,558,371 

1880 50,155,783 

1890 62,947,714 

1900 75,994,575 

1910 91,972,266 

1920 105,710,620 


GRAPHICS 


105 


A  glance  at  the  diagram  gives  one  a  pretty  clear  idea  regarding 
the  increase  in  numbers. 

2,  Show  also  the  percentage  of  increase  in  each  decade  as  given 


in  the  table  below: 

Year 

Per  Cent 

Year 

Per  Cent 

Year 

Per  Cent 

1800. . 

....35.1 

1850.. 

....35.9 

1900.. 

....20.7 

1810. . 

....36.4 

1860. . 

....35.6 

1910. . 

....21.0 

1820. . 

....33.1 

1870. . 

....26.6 

1920. . 

....14.9 

1830. . 

....33.5 

1880. . 

....26.0 

1840. . 

....32.7 

1890. . 

....24.9 

78.  Line  diagrams.  As  we  are  especially  interested  in 
the  way  quantities  change  from  time  to  time,  it  is  the  ends 
of  the  bars  that  hold  our  attention.  Such  changes  are 
more  easily  seen  if  we  plot  merely  the  ends  of  the  bars  and 
then  connect  these  ends  by  a  series  of  straight  lines  or  a 
smooth  curve  as  in  Fig.  14  (obtained  from  table  of  Fig.  13) 
45%t  M  I  M  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  n        and  Fig.  15  (number  of  pupils 

enrolled  in  a  certain  school). 


Fig.  14  Fig.  15 

Such  figures  are  called  graphs.  A  point  is  plotted  to 
correspond  to  each  pair  of  numbers  in  the  table,  and  these 
points  connected  by  a  straight  line  or  a  smooth  curve. 

8 


106 


BEGINNERS'  ALGEBRA 


EXERCISES 

1.  Draw  the  graph  for  the  following  table  showing  the  average 
weight  of  boys  compared  with  their  height: 


Height 
in  In. 

Weight 
in  Lbs. 

Height 
in  In. 

Weight 
in  Lbs. 

Height 
in  In. 

Weight 
in  Lbs. 

Height 
in  In. 

Xlt^. 

35.. 

.32.0 

43.. 

.43.5 

50.. 

.59.5 

57.. 

.  83.5 

36.. 

.33.5 

44.. 

.45.5 

51  .t 

.63.0 

58.. 

.  87.5 

37.. 

.34.5 

45.. 

..47.5 

52.. 

.66.0 

59.. 

.   91.5 

38.. 

.36 

46.. 

..49.5 

53.. 

.69.0 

60.. 

.  95.0 

39.. 

.37.5 

47.. 

..51.5 

54.. 

.72.5 

61.. 

.  99.5 

40.. 

.39 

48.. 

..53.5 

55.. 

.75.5 

62.. 

.105.0 

41.. 

.40.5 

49.. 

..55.5 

56.. 

.79.5 

63.. 

..109.5 

42.. 

..42.0 

2.  Draw  the  temperature  graph  from  the  reading  for  a  certain 
day  in  June.  Lay  off  the  time  scale  horizontally  to  the  right  from 
the  lower  left-hand  comer  of  the  ruled  paper,  using  -^  of  an  inch  to 
each  hour.  Lay  off  the  temperature  scale  vertically  on  the  left, 
using  j^  of  an  inch  to  a  degree. 


Hourly  Temperature  Reading 


MIDNIGHT  .  . 

..77° 

8:00  a.m... 

..81° 

3:00  p.m... 

..83° 

1:00  a.m... 

..75° 

9:00  a.m... 

..84° 

4:00  P.M. . 

..68° 

2:00  a.m... 

..74° 

10:00  a.m... 

..81° 

5:00  p.m.. 

..70° 

3:00  a.m... 

..73° 

11:00  a.m... 

..88° 

6:00  p.m.  . 

..76° 

4:00  a.m... 

..73° 

NOON 

...91° 

7:00  p.m.. 

..76° 

5:00  a.m... 

..72° 

1:00  p.m... 

...92° 

8:00  p.m.. 

..76° 

6:00  A.M... 

..75° 

2:00  p.m... 

...92° 

9:00  p.m.. 

..76° 

7:00  a.m... 

..76° 

Note  the  sudden  drop  in  temperature  between  2:00  and  5:00 
o'clock.    There  was  a  severe  storm  at  that  time. 

3.  Contrast  the  graphs  just  drawn  with  that  of  the  coldest  day 
the  next  winter: 

Temperature  Readings  for  December  25,  26,  1903 
Dec.  25: 

6:00  A.M 18°        NOON 28°         6:00  p.m 4** 

8:00  A.M 18°         2:00  P.M 12°         8:00  p.m 0* 

10:00  A.M 21°         4:00  P.M 8°       10:00  p.m..  . . -2° 


GRAPHICS 


107 


-5° 

10:00  a.m.... 

.   0° 

4:00  P.M. 

..   14° 

-r 

NOON 

.  5° 

6:00  p.m. 

..   16° 

-6° 

2:00  p.m.... 

.12** 

8:00  p.m. 

..   18° 

iperati 

lire  graph  for 

your 

own  locality. 

Get  the 

Dec.  26: 

midnight  .  .  . 
6:00  a.m... 
8:00  a.m... 

4.  Plot  a  te: 
data  for  several  days  from  the  newspaper  weather  reports. 

5.  The  following  table  gives  xhe  records  for  height  in  the  pole 
vault  for  the  United  States  and  the  years  when  made.  Display 
these  on  a  diagram. 

Year  Height 


Year  Height 

1892 11'  5|" 

1898 inoV' 

1904 12'  lY 


1906 12'  4|" 

1907 12'  5^" 

1908 12'  W 


Year  Height 

1910 12' 101" 

1912 

1919 13'   3 


13'   2i" 


T¥ 


6.  Plot  the  following  table  showing  the  height  above  sea  level  of 
points  on  the  Rock  Island  Railroad  between  Chicago  and  Rock 
Island: 


Miles 

Height  in  Feet 

Station 

Miles 

Height  in  Feet 

Station 

0 
30 
40 
51 

85 

603 
716 
538 
612 

484 

Chicago 

Mokena 

Joliet 

Minooka 

Ottawa 

114 
137 
159 
181 

466 
670 
641 
570 

Bureau 
Sheffield 
Geneseo 
Rock  Island 

7.  Draw  a  graph  from  the  following  table  showing  the  retail 
price  of  eggs  per  dozen: 

Cents  Cents  Cents  Cents 

1913 33      1915 31      1917 46     1919 60 

1914 33     1916 36      1918 54     1920 64 

8.  Plot  profile  of  the  bottom  of  a  river  from  the  following  table 
of  soimdings  made  at  points  a  hundred  feet  apart  straight  across. 

0.0  10.1  9.4  7.4  2.8 

8.1  10.3  9.5  6.5  1.4 

10.6  10.5  9.2  5.3  0.8 

10.3  10.1  8.2  3.9  00 

A  sounding  is  the  depth  of  the  water  found  by  dropping  into  the 
water  a  chimk  of  lead  fastened  to  a  string. 

Where  is  the  channel  of  the  river?    Answer  from  the  graph, 


108 


BEGINNERS*  ALGEBRA 


79.  Comparison  of  graphs.  It  is  sometimes  of  interest  to 
compare  two  graphs.  This  may  be  done  more  easily  if  the 
graphs  are  drawn  on  the  same  diagram.  To  prevent  con- 
fusion, the  graphs  may  be  distinguished  by  the  use  of  different 
colors  or  different  kinds  of  lines,  such  as: 


The  figure  below  shows  the  growth  of  the  population  of 
the  cities  of  New  York,  Chicago,  and  Philadelphia: 


thousands 


1300:: 
1400  [I 
1300  ':[ 

: 

lilMiiii 

m 

J200 : : 
1100 ;; 



Mill 

::x:- 

::|::J::;:::;::: 

1000 ;: 
900 :; 
800 :: 

-  -- 

:;l:E::::i::::::; 

f  j  -  ■- ^ —  •■■ 

700  :: 

600:: 
5oo:; 
400 :: 

■--■:: 

MfiMi 

--^H'''W'' 

300:: 

0/\A       ' 

-  .      •  -      T 

.:::: 

liiBi 

-::::;:::;:::;:;; 

100 :; 

-.:.::. i 

§  i 


i  I 


Fig.  16 


What  does  the  steepness  of  the  graph  indicate? 
Which  of  the  three  cities  was  growing  the  most  rapidly  from 
1880  to  1890? 
Which  had  the  slowest  growth? 


GRAPHICS 


109 


EXERCISES 


1,  Compare  the  enrollment  in  two  different  departments  of  a 
certain  school  for  successive  years. 


A 

A 

A 

B 

B 

B 

210 

191 

54 

192 

142 

180 

207 

164 

77 

193 

125 

202 

216 

168 

55 

184 

109 

243 

186 

27 

72 

173 

103 

285 

178 

38 

84 

184 

138 

318 

161 

37 

104 

147 

157 

337 

2.  Plot  on  one  diagram  graphs  of  the  average  prices  given  in  table : 


Years 

Wheat 

Oats 

Com 

Potatoes 

1899 

.58 

.24 

.30 

.39 

1904 

.93 

.31 

.44. 

.45 

1909 

.99 

.40 

.60 

.54 

1912 

.76 

.32 

.49 

.50 

1913 

.80 

.39 

.69 

.69 

1914 

.89 

.44 

.64 

.49 

3.  Draw  on  the  same  diagram  graphs  showing  the  imports  and 
exports  of  the  United  States  for  the  following  years: 


Imports 

Exports 

Imports 
Millions 

Exports 
Millions 

Year 

Millions 

Millions 

Year 

of  Dollars 

of  Dollars 

of  Dollars 

of  Dollars 

1900 

850 

1371 

1908 

1194 

1835 

1901 

823 

1460 

1909 

1312 

1638 

1902 

903 

1355 

1910 

1557 

1710 

1903 

1026 

1392 

1911 

1527 

2014 

1904 

991 

1435 

1912 

1653 

2170 

1905 

1118 

1492 

1913 

1813 

2429 

1906 

1227 

1718 

1914 

1893 

2330 

1907 

1434 

1854 

80.  Graphs  mechanically  drawn.  The  temperature  read- 
ings given  in  several  exercises  of  Art.  78  were  obtained  by 
readings  of  the  thermometer  every  hour.  If  readings  had 
been  taken  more  frequently —  say  every  fifteen  minutes — the 
plotted  points  would  have  been  closer  and  the  graph  would 
indicate  more  accurately  the  actual  changes  in  temperature. 


110 


BEGINNERS'  ALGEBRA 


An  instrument,  called  the  thermograph,  has  been  invented 
to  draw  a  temperature  graph  automatically .     A  pe  n  attached 

to  a  metal  ther- 
mometer draws 
a  continuous 
line  on  a  sheet 
of  paper  rolled 
on  a  cylinder 
which  is  kept 
revolving  at  a 
constant  rate. 
The  thermo- 
Pjq  17  graph   and    the 

line  drawn  by 
it  are  shown  in  Figs.  17  and  18.  The  rise  and  f-all  of  the 
graph  denote  the  rise  and  fall  of  the   temperature.     We 


7  8 

XII2468  10M2  46  8  10X11  2468  lOM  2  4  68  10X1124  6  8  10  M'2  4  6  8  10  XII2  4  6  8  10>»2  4 


6 


ggggim 


Fig.  18 


can   read  the  temperature  at  any  moment  directly  from 
the  graph  by  finding  the  time  on  the  time  scale  and  reading 


GRAPHICS 


111 


off  the  height  of  the  graph  at  that  point.      For  instance,  at 
4:00  P.M.  October  6  the  temperature  was  76°. 

EXERCISE 

1.  From  the  temperature  graph  in  Fig.  18  read  the  temperature 
at  noon  October  5,  at  noon  October  6,  at  6:00  P.M.  October  7,  at 
6:00  A.M.  October  8. 

When  was  the  temperature  78°?    32°? 
What  hours  during  the  day  was  it  warmest? 
What  hours  during  the  day  was  it  coldest? 

81.  Graphs  used  for  reckoning.  Make  a  table  of  the 
prices  of  different  lengths  of  gingham — say  from  1  to  10 
yards  at  15  cents  a  yard — and  plot  the  graph  (Fig.  19). 
Lay  off  the  scale  of  yards  along  a  horizontal  line  or  axis  and 
the  scale  of  costs  along  a  vertical  linear  axis. 


Yards 

Cents 

cts. 

1 

15 

120 

2 

.....30 

110 

3 

4 

45 

60 

100 

5 

75 

90 

6 

7 

....  90 
....105 

80 

8 

....  120 

70 

9..... 
10. .... 



60 

What    is    peculiar 

50 

about   these   points? 

40 

Plot  the  price  of  5J 

30 

yards;    of 

3J  yards. 

?0 

Taking  it  for  grant- 

ed that  the  graph  is 

10 

a  straight 

line,  read 

directly  from  the 

graph  the 

price  of  8 

8  y^s 


yards;  the  price  of  7^  yards;  the  price  of  2f  yards. 


112  BEGINNERS'  ALGEBRA 

How  many  yards  can  be  bought  for  50  cents?  Read  the 
number  of  yards  from  the  graph  and  then  test  your  result 
by  computing  the  amount. 

This  graph  differs  very  materially  from  those  we  have 
been  considering  (excepting  the  automatically  drawn 
graphs).  In  the  graphs  hitherto  drawn  only  those  points 
that  were  actually  plotted  had  a  definite  value.  The  Hnes 
joining  them  were  merely  to  aid  the  eye.  In  this  price 
graph  every  point  on  the  graph  has  a  meaning.  It  corre- 
sponds to  a  certain  number  of  yards  at  a  certain  price. 
This  graph  shows  the  relation  between  the  price  and  the 
number  of  yards  of  goods  in  the  form  of  a  picture.  The 
same  relation  may  be  shown  algebraically  by  means  of  a 
formula: 

Price  =  15  X  number  of  yards 

Results  may  be  read  off  from  such  graphs  very  readily. 
Graphs  of  this  kind  are  used  very  extensively  as  ready 
reckoners.  A  few  illustrations  are  given  in  the  exercises 
below. 

EXERCISES 

Draw  graphs  and  give  formula  for  each. 

1.  Garden  hose  sells  at  17  cents  a  foot.  Draw  a  sales  graph  and 
read  from  the  graph  the  price  of  25  feet,  33  feet,  and  12J  feet. 
How  many  feet  can  be  purchased  for  $1.50?  Write  down  an 
algebraic  formula  and  test  your  results  by  computation. 

2.  Tenpenny  nails  sell  at  7  cents  a  pound.  Draw  a  sales  graph. 
How  many  pounds  will  25  cents  buy? 

3.  Draw  a  graph  that  can  be  used  in  filling  out  an  order  for  any 
number  of  a  certain  kind  of  bolt  by  weighing,  allowing  18  bolts 
to  the  pound. 

4.  Cold-storage  eggs  sell  at  50  cents  a  dozen,  and  fresh  eggs 
at  60  cents  a  dozen.  Make  a  sales  graph  for  each,  putting  the 
two  graphs  on  the  same  paper  and  on  the  same  axes.     Find  the 


GRAPHICS  113 

answers  to  the  following  questions  on  the  graphs:  How  many 
more  cold-storage  eggs  can  you  get  for  $4.50  than  fresh 
eggs?  What  is  the  difference  in  the  price  of  3  dozen  eggs  of 
each  kind? 

5.  Water  is  flowing  into  a  tank  at  the  rate  of  2  gallons  a  second. 
Draw  a  graph  showing  the  number  of  gallons  of  water  in  the  tank 
at  any  time  after  the  water  is  turned  on. 

6.  Water  is  flowing  into  a  tank  from  two  pipes — from  one  at 
the  rate  of  2  gallons  a  second,  from  the  other  at  the  rate  of  3  gallons 
a  second.  Draw  graphs  showing  the  nimiber  of  gallons  of  water 
in  the  tank  at  any  time  after  the  water  is  turned  on.  Answer  the 
following  questions  from  the  graphs:  How  much  water  flows  in 
from  each  pipe  in  10  seconds?  From  both  pipes?  How  much 
more  flows  in  from  the  larger  than  from  the  smaller  in  15  seconds? 
How  long  will  it  take  10  gallons  to  flow  in  from  each  pipe?  If  the 
tank  holds  25  gallons,  how  much  longer  will  it  take  the  smaller  pipe 
to  fill  it  than  the  larger? 

7.  An  automobile  is  moving  at  the  rate  of  20  miles  an  hour. 
Draw  a  graph  showing  the  distance  traveled  in  any  given  time 
less  than  3  hours. 

8.  One  automobile  is  moving  at  the  rate  of  20  miles  an  hour, 
another  at  the  rate  of  25  miles  an  hour.  Draw  graphs  showing 
the  distances  traveled  at  any  given  time.  How  far  apart  will  the 
machines  be  in  8  hours?  How  long  will  it  take  each  machine  to 
go  65  miles? 

9.  Draw  a  graph  that  can  be  used  to  change  inches  into  centi- 
meters.    1  inch  =  2. 5  cm.  (about). 

10.  Draw  a  graph  that  can  be  used  to  change  kilometers  to  miles. 
Show  by  means  of  this  graph  the  speed  in  miles  per  hour  of  a 
French  train  that  is  reported  as  having  a  speed  of  90  kilometers 
per  hour.     1  mile  =1.609  kilometers. 

11.  Draw  a  graph  to  be  used  in  changing  cubic  inches  into  gal- 
lons, a  gallon  being  231  cubic  inches. 

12.  Draw  an  interest  graph  for  the  interest  on  one  dollar  at  5  per 
cent  for  a  number  of  years. 


114 


BEGINNERS'  ALGEBRA 


13.  A  saves  $5  a  week.  B  has  $10  given  to  him  and  then  starts 
saving  at  the  rate  of  $4  a  week.  Draw  graphs  showing  how  much 
money  each  one  has  at  the  end  of  a  given  number  of  weeks.  When 
will  they  have  the  same  amount? 

14.  The  cost  of  setting  up  the  type  for  printing  a  certain  circular 
is  two  dollars.  The  printer  charges  fifty  cents  per  hundred  for 
copies.  Draw  a  graph  which  could  be  used  for  finding  the  total 
cost  of  any  number  of  circulars. 

15.  If  the  value  of  a  house  costing  $5,000  decreases  $200 
each  year,  draw  a  graph  showing  what  the  house  is  worth  at  any 
time. 

16.  A  saves  at  the  rate  of  $3  a  week.  After  3  weeks  B  begins  to 
save  at  the  rate  of  $4  a  week.  When  will  they  have  saved  the  same 
amount?    Answer  from  graphs. 

17.  The  schedule  of  a  train  running  from  town  A  to  town  E  is 

given  in  tabular  form  on  the  left  below,  and  in  graphical  form  in  the 

figure.  ^ 

|'^i|es  1  2   hours 

rt  60 


Miles 

Station 

Daily 

0 

A 

12  Noon 

20 

B 

arr 

12:40 

Ive 

12:50 

24 

C 

1:10 

44 

D 

arr. 

1:30 

Ive. 

1:35 

60 

E 

1:40 

A- 


hours 


Fig.  20 


The  graph  is  worth  careful  study.  What  do  the  horizontal  lines 
mean?  Between  what  points  was  the  train  traveling  at  the  fastest 
rate?  At  the  slowest?  What  was  the  average  rate  between  A 
and  E,  including  stops? 

It  is  a  common  practice  in  railroad  offices  to  chart  the  movement 
of  trains  in  this  way.  Many  trains  are  charted  on  the  same 
paper. 


GRAPHICS 


115 


The  schedule  of  a  train  leaving  E  at  1 :  00  p.m.  and  running  to  A 
in  H  hours  without  stop  would  be  indicated  by  a  line  drawn  from 
near  the  top  of  the  diagram  down  to  the  right.     Draw  the  graph. 

18.  Chart  the  following  train  schedules  on  one  chart. 


Miles 

Station 

1 

Daily 

2 

Daily 

0 

A 

Ive. 

1:00  p.m. 

1:30  p.m. 

6 

B 

16 

C 

arr. 
Ive. 

1:50 
2:00 

1:55 

30 

D 

arr. 
Ive. 

2:40 
2:55 

40 

E 

3:20 

2:30     arr. 

2:35     Ive. 

46 

F 

2:50 

82.  Problems  solved  graphically.  Many  problems  can 
be  solved  by  graphs  without  any  use  of  equations.  Espe- 
cially is  this  true  of  problems  concerning  bodies  moving  at  a 
constant  speed,  such  as  an  automobile  moving  at  the  rate 
of  15  miles  an  hour  or  train  at  the  rate  of  50  miles  an  hour. 
In  such  cases  we  think  of  the  object  as  moving  through 
equal  distances  in  equal  times.  The  last  two  exercises  in 
the  last  article  deal  with  this  kind  of  motion. 

EXERCISES 

1.  A  and  B  start  at  the  same  place  at  the  same  time  and  travel 
in  the  same  direction,  B  at  ^q 
the  rate  of  10  miles  an  hour 
and  A  at  the  rate  of  12 
miles  an  hour.  How  far 
apart  are  they  in  5  hours? 
When  were  they  8  miles 
apart? 

Let    the    pupil    answer 

the  question  from  the  graph 

at  the  right  (Fig.  21).  __  ^  

"^  •  "  ''  4  hours  5 


116 


BEGINNERS'  ALGEBRA 


2.  A  and  B  start  from  the  same  place  and  travel  in  the  same 
direction.  If  A  travels  at  the  rate  of  6  miles  an  hour  and  B  at  the 
rate  of  9  miles  an  hour,  after  how  many  hours  will  they  be  6  miles 
apart?    Solve  by  graph. 

3.  A  travels  at  the  rate  of  3  miles  an  hour.  After  4  hours  B, 
who  started  at  the  same  time  and  place,  is  15  miles  ahead  of  A. 
Find  by  graph  the  rate  at  which  B  travels.  An  approximate  answer 
is  sufficient. 

Solve  the  three  preceding  examples  by  arithmetic. 

4.  A  and  B  start  from  the  same  place  at  the  same  time  and 
travel  in  opposite  directions.  If  A  travels  at  the  rate  of  5  miles  an 
hour  and  B  travels  at  the  rate  of  7  miles  an  hour,  how  far  apart 
will  they  be  in  6  hours?  At  what  time  will  they  be  18  miles 
apart? 

5.  A  and  B  start  from  the  same  place  and  travel  in  the  same 
direction.  A  travels  at  the  rate  of  8  miles  an  hour.  Two  hours 
after  A  has  left,  B  starts  after  him  at  the  rate  of  12  miles  an  hour. 
When  will  B  overtake  A?    (See  Fig.  22.) 

B  6.  An  express  train  leaves  a  station 

two  hours  after  a  freight  traveling 
in  the  same  direction.  The  freight 
is  running  at  the  rate  of  20  miles  an 
hour,  the  express  at  the  rate  of  45 
miles  an  hour.  In  how  many  hours 
will  the  express  overtake  the  freight? 

7.  A  is  walking  at  the  rate  of  3^ 

miles  an  hour.      After  4  hours  B 

2    3   4  hours  starts  after  him  on  his  bicycle  at 

Fig.  22  the  rate  of  15  miles  an  hour.    When 

will  B  overtake  A? 


:  = 

:::::; 

■  -  ■  ■  -H 

tr-z 

1 

=; 

-J 



1 

— 

:; 

¥ 

Wm 

P^'^ 

i[ 

:,' 

:i    ::::: 

:  ::: 

:\\ 

8.  A  rode  away  from  town  on  his  wheel.  Two  hours  after  he 
left,  B  started  after  him  in  his  automobile  at  the  rate  of  40  miles 
an  hour  and  overtook  him  in  half  an  hour.  How  fast  was  A  riding? 
What  distance  had  he  covered? 


GRAPHICS 


117 


9.  A  and  B  start  at  the  same  time  and  travel  east.  A  starts 
froni  a  place  12  miles  west  of  the  place  from  which  B  starts  and 
travels  at  the  rate  of  20  miles  an  hour.  If  B  is  going  at  the  rate  of 
12  miles  an  hour,  when  will  A  overtake  B? 

10.  A  and  B  start  at 
the  same  time  from 
places  70  miles  apart 
and  travel  toward  each 
other,  A  at  the  rate  of 
20  miles  an  hour  and  B 
at  the  rate  of  15  miles 
an  hour.  When  and 
where  will  they  meet? 
(See  Fig.   23.) 

Let  the  pupil  read  the 
answers  from  the 
graphs. 

83.  Graphs  of  al- 
gebraic   expressions. 

Any  algebraic  expres- 
sion in  one  unknown  can  be  represented  graphically.     As 
an  illustration  take  the  expression 

Now  %  may  be  any  number.  If  %  is  given  any  particular 
value,  say  5,  then  %-2  will  become  5-2,  or  3.  Now,  by 
giving  %  a  number  of  values  in  succession  and  finding  the 
corresponding  value  of  rj;— 2  in  each  case,  we  can  build  up  a 
table  of  pairs  of  values  for  %  and  %—2. 

If  x  =  2  then  x-2  =  ^ 
%=Z  x-2  =  \ 

x  =  4:  a;-2  =  2 

iv=6  a;-2  =  4 

x  =  9  x-2  =  7 

x  =  Yl  r^-2  =  10 

Because  the  expressions  x  and  x—2  can  have  various 
values,  they  are  called  variables. 


118 


BEGINNERS'  ALGEBRA 


These  pairs  of  values  may  be  plotted,  each  pair  giving  one 
point  on  the  graph.  Use  the  horizontal  scale  for  the  x 
values  and  the  vertical  scale  for  the  x—2  values. 

The  plotted  points  all 
seem  to  lie  on  a  straight 
Hne  NP  (Fig.  24).  Com- 
pute the  value  oi  x—2  for 
some  value  of  x  between  the 
values  plotted,  say  x  =  3^ 
and  x  =  5y  and  plot. 

Two  important  facts  are 
to  be  noticed  here: 

(1)  Any  set  of  values  of 
the   two   variables   x  and 


TL 

2 

^ 

7 

.     _^7 

lU  --   - 

t^ 

_    ^ 

4^ 

?' 

•>        --^2 

■                 ^?0 

z^^ 

-(X 

-6? 

^-4?^-         - 

u     %  5 

10 

■X 


Fig.  24 

x—2  determine  a  point  on  the  line  NP, 

(2)  For  any  point  on  the  line  NP  there  are  two  numbers, 
one  from  each  scale,  that  fit  the  variables  x  and  x—2. 

For  instance,  the  scale  readings  for  the  point  Q  are: 

horizontal,  7^     ^,    ^  .       Sl  —  x 

^.    .     '     >    that  IS,    <  -  ^ 

vertical,      5  J  '     15  =  x—2 

The  scale  readings  for  a  point  are  called  the  coordinates  of 
the  point.  They  are  often  written  in  the  form  (7,  5) ;  the 
horizontal  coordinate  is  written  first. 

From  the  graph  find: 

(1)  values  oi  x—2  when  51;  =  8,  10,  11 

(2)  values  of  x  when  :x;-2  =  5,  3,  9,  13 

The  pairs  of  values  may  be  extended  by  taking  either 
smaller  or  larger  values  of  x. 

If  ^  =  1  thenrx:-2=-l 
li  x==  S  then  x-2=  -5 

We  may  plot  the  pairs  (1,  —1)  and  (  —  3,  —5)  by  extend- 
ing the  scales,  the  x  scale  to  the  left  and  the  x—2  scale 


GRAPHICS 


119 


downward.  The  point  for  the  pair  (1,  —1)  then  falls  one 
unit  below  the  +1  on  the  %  scale.  The  point  for  (  —  3,  —5) 
falls  five  units  below  the  —  3 
on  the  %  scale.     (Fig.  25.) 

Positive  numbers  are  to 
be  laid  off  to  the  right  of 
the  vertical  axis  or  above 
the  horizontal  axis. 

Negative  numbers  are  to 
be  laid  off  to  the  left  or 
downward. 

The  straight  Une  L  is  the 
graph  of  the  expression  %—2. 


Fig.  25 


EXERCISES 

I.  Read  the  coordinates  for  the  points  marked  in  Fig.  26. 
2.  Plot     the     following 

points.  The  horizontal  co- 
ordinate is  given  first.  (3, 
4),  (-2,  3),  (2,  -3),  (-3, 

-5),  (0,  0),  (3,  0),  (0,  5), 
(-2,    1),    (-5,     -4),    (0, 

-5),    (-7,   0). 

3.  Find  the  values  of 
x-5  for  x  =  %,  7,  6,  5,4,3, 
2,1,0,   -1,   -2,   -3,  -4, 

-5,  -6,  -7,  -8,  and  plot 
on  cross-section  paper. 

4.  Find  the  values  oi  Z—%  for  integral  values  of  x  between 
a:=  — 6  and  :»j=  +6  and  plot. 

5.  Plot2:x:.  6.  Plot2:x;-3. 
7.  Plot4-3:x:.                           8.  Plot  -2:c+5. 
9.  Plot  -2a; -3.                     10.  Plotic+S. 

II.  What  kind  of  line  do  you  think  the  graph  of  an  expression 
of  the  form  ax-\-h  is  where  a  and  h  are  supposed  to  be  special  known 
numbers? 


T  T 

...a:  j^_ 

^^      -5-       ., 

^■AV 

J 

5^ .^ 

-.-i--^^-.-^^^-_- 

5v-         Uv 

^     ~(^i- 

± 

Fig.  26 


120 


BEGINNERS'  ALGEBRA 


84.  Graphs  of  equations.    The  exercises  of  the  last  article 
may  be  put  in  a  slightly  different  form. 

For  convenience  we  may  denote  the  second  of  the  t-^o 
variables  x  and  x+S  by  a  single  letter,  thus: 

y=x+S 
giving  a  letter  for  each  variable.  It  is  to  be  imderstood  that 
the  y  stands  for  x+3  in  this  case.  The  graph,  of  course, 
will  be  exactly  the  same  as  before.  We  assign  values  to  x 
and  find  the  corresponding  value  of  y  and  plot  as  before, 
with  the  horizontal  line  as  the  jc-axis  and  the  vertical  line 
as  the  y-axis. 

The  equation  y  =  x-\-S  may  be  put  in  the  form  y—x  =  S, 
but  the  graph  will  not  be  changed. 

The  fact  that  the  difference  between  two  nimibers  equals 
3  may  be  stated  in  two  ways: 


Algebraic  Statement 

y-x=S 


Graphic  Statement 

/ 

/ 

/ 

/ 

/ 

/ 

0 

/ 

Fig.  27 


In  general,  all  equations  in  two  variables  may  be  repre- 
sented graphically.  There  are  many  different  kinds  of  equa- 
tions. Each  kind  has  its  own  peculiar  graph.  Some  of  these 
graphs  are  very  simple,  and  others  are  rather  complicated. 
But  each  displays  the  characteristics  of  its  equation  in  very 
striking  manner.  It  was  a  great  step  in  mathematics  when 
this  relation  between  equations  and  lines  was  found.  We 
owe  the  idea  to  a  Frenchman  named  Descartes  who  lived 
from  1596  to  1650. 

Illustration.     Draw  the  graph  of  the  ^equation  2x-\-y  =  10. 


GRAPHICS 


121 


Values  are  assigned  to  x  and  the  corresponding  values  of 
y  found.     (Fig.  28.) 

y 


If  x  =  l      2+y  =  l0  . 

■.   y  =  8 

x  =  Z      6+>'  =  10 

y=4 

x  =  ^     10+>'  =  10 

y  =  Q 

Plot  the  pairs 

(1.8) 

(3.4) 

(5.0) 

II^IIII     11= 

^ 

^ 

__.i            _ 

:i«  itiiii 

-    4       \ 

-.-.-r.-A'-.z 

0          2        4    V6 

Fig.  28 


EXERCISES 

Draw  a  graph  for  the  following  equations: 

1.  y  =  2x-{-4:  2.  y+4aj=4 

3.  y+4rc+4=0  4.  2y  =  Sx 

85.  Solving  equations  for  one  variable.  The  work  of 
graphing  an  equation  is  usually  made  simple  if  the  equation 
is  first  solved  for  one  of  the  unknowns  in  terms  of  the  other. 

Consider  the  equation 

2x-{-y  =  9 
Untangle  the  y  and  get 

y  =  9-2x 

and  then  find  the  value  of  9— 2^1;  for  various  values  of  x. 
Why  is  this  any  better  than  the  way  given  in  the  last  article  ? 

In  assigning  values  to  x  it  is  better  to  select  numbers  in 
regular  order  rather  than  to  choose  them  hit  or  miss. 

9 


122  BEGINNERS'  ALGEBRA 

PROBLEMS 
Solve  these  equations  for  y  in  terms  of  x  and  then  plot  the  graph. 

1.  2jc+33;  =  6  2.  2%-Zy  =  ^ 

3.  5y4-2a:  =  0  4.  3^-4y=12 

'  GENERAL   EXERCISES 

1.  The  difference  between  two  numbers  is  12.  Write  the  equa- 
tion and  draw  the  graph. 

2.  The  quotient  of  two  nimibers  is  4.  Write  the  equation  and 
draw  the  graph. 

3.  Twice  a  number  plus  a  second  number  plus  4  is  always  zero. 
Write  the  equation  and  plot  the  graph. 

4.  If  one  side  of  a  rectangle  is  always  3  less  than  twice  the  other 
side,  write  the  equation  and  draw  the  graph  showing  the  relation 
between  the  sides  of  all  rectangles  filling  the  requirement. 

5.  The  value  of  a  number  of  nickels  is  always  3  times  the  value 
of  a  number  of  one-cent  pieces.  Draw  the  graph  showing  the  rela- 
tion between  the  number  of  nickels  and  the  number  of  one-cent 
pieces. 

6.  The  value  of  a  ntimber  of  nickels  is  always  2  less  than  3 
times  the  value  of  a  number  of  one-cent  pieces.  Draw  the  graph 
showing  the  relation  between  the  number  of  nickels  and  the  number 
of  one-cent  pieces. 

7.  In  five  years  A  will  be  twice  as  old  as  B.  Draw  the  graph 
showing  the  relation  between  their  ages. 

8.  The  sum  of  the  two  digits  of  a  number  is  12.  Draw  a  graph 
showing  the  relation  between  the  digits. 

9.  A  man  can  row  one  mile  up  a  stream  against  a  current  in 
25  minutes.  Draw  a  graph  showing  the  relation  between  the  rate 
of  the  current  and  the  rate  the  man  can  row  in  still  water. 

10.  On  the  same  diagram  draw  the  graphs  of  x-\-y  =  2,  3ac— ^  =  6, 
Zy-x  =  ^. 

11.  Draw  a  graph  showing  the  position  of  all  points  that  are 
twice  as  far  from  the  y-axis  as  they  are  from  the  ic-axis. 


CHAPTER  VI 

Linear  Equations  in  Two  Unknowns 

86.  Graphic  solutions.  Let  us  consider  the  following 
problem.  What  are  the  numbers  whose  sum  is  4?  The 
question  may  be  stated  algebraically  or  graphically : 

Graphic  Statement 

.    y 


Algebraic  Statement 
(1)  x^y^^ 


\ 

s 

«^^ 

s 

s 

s 

s. 

( 

7 

s 

s. 

s 

_ 

Fig.  29 

There  are  two  ways  of  getting  an  answer  to  the  question : 

(1)  Any  pair  of  values  of  x  and  y  that  satisfy  the  equa- 
tion may  serve  as  an  answer  to  the  problem,  for  instance, 
1  and  3,  or  — 6  and  10.  Find  four  other  sets  of  answers 
from  the  equation. 

(2)  The  coordinates  of  any  point  on  line  (1)  may  serve  as 
answers,  for  instance,  the  points  that  give  2  and  2,  or— 2 
and  6.  Find  from  the  graph  four  other  sets  of  answers  to 
the  question. 

-  There  is  an  unlimited  number  of  answers  to  the  question. 
Any  pair  of  values  that  satisfies  the  equation  is  called  a 
solution  of  the  equation. 

But   suppose   another   fact   is   known   about   these  two 

123 


124 


BEGINNERS'  ALGEBRA 


numbers.  Suppose  the  question  reads:  What  are  the 
numbers  whose  sum  is  4  and  whose  difference  is  1? 
(Fig.  30.) 

Graphic  Statement 

y 


Algebraic  Statement 

(1)  x+y  =  4. 

(2)  x-y  =  l 


\ 

\ 

/ 

s 

/ 

s 

/ 

X 

/ 

\, 

0 

/ 

s 

/ 

s 

/ 

l'i\ 

/ 

W 

Fig.  30 

The  coordinates  of  any  point  on  line  (1)  will  satisfy  the 
sum  condition;  the  coordinates  of  any  point  on  line  (2) 
will  satisfy  the  difference  condition.  As  there  is  but  one 
point  common  to  the  two  lines,  there  is  but  one  pair  of 
values  that  will  satisfy  both  conditions  of  the  problem.  If 
the  graphs  have  been  carefully  drawn,  the  coordinates  of 
this  common  point  may  be  read  from  the  scales  on  the  axes 
with  some  degree  of  accuracy.  To  check  the  result  substi- 
tute the  numbers  in  the  two  equations. 

Reading  the  coordinates  of  the  common  point  of  the  case 
in  hand,  we  get 

These  check  when  substituted  in  the  equations 

x-{-y  =  4:  x—y  =  l 


2i  +  U  =  4 


2*-U  =  l 


We  have  thus  solved  the  problem  graphically.     The  alge- 
braic solution  will  be  given  later. 

87.  Short  method  of  drawing  graph  of  a  linear  equation. 

Any  equation  of  the  form 

ax+by=c 


LINEAR  EQUATIONS   IN  TWO  UNKNOWNS       125 


such  as  Sx-\-5y=lS 

2x-3y  =  5 
x-7y=-S 

is  called  a  linear  equation  because  its  graph  is  a  straight 
line.  The  x  and  y  are  the  two  unknowns,  while  a,  b,  and  c 
are  any  known  numbers  whatsoever. 

Only  one  straight  Hne  can  be  drawn  through  two  points. 
The  graph  of  a  linear  equation  is  a  straight  line.  Conse- 
quently only  two  points  are  needed  for  drawing  the  graph. 

y 

To  find  the  coordinates  of  two 
points,  assign  to  one  of  the  letters, 
say  X,  any  two  values  and  find 
the  corresponding  values  of  y. 
(Fig.  31.) 

For  instance ,  Sx-\-2y  =  9 

Put  rjc  =  1 ,  find  ;y  =  3 

Put  x  =  5,  find  y=  —3 

Fig.  31 

It  is  well  to  choose  points  that  are  rather  far  apart. 
Why? 

The  points  where  the  line  crosses  the  axes  are  often  very 
satisfactory  points  to  choose.  The  x  of  the  point  where  the 
line  crosses  the  y-axis  must  be  zero.  Why?  To  find  the 
coordinates  of  the  y-axis  crossing,  put  x  =  0  and  find  the 
corresponding   value  of   y.  y 

(Fig.  32.) 


\ 

^ 

5^3 

^ 

0 

"T 

w 

-s     ^^ 

~^*  ^r 

Thus, 
For  X- 


0 


2^-3:V=12 
y=  —4 


What  must  be  the  value 
of  y  for  the  point  where  the 
line  crosses  the  :>^-axis  ?  Find 
the  corresponding  value  of  x. 


4 

/ 

0 

^ 

e, 

0 

/ 

4 

»^ 

V 

> 

n 

J" 

4 

_ 

^ 

_ 

- 

J 

Fig.  32 


126  BEGINNERS'  ALGEBRA 

EXERCISES 

Solve  by  the  graphic  method: 

1.  x+2y=l  2.  x+y  =  l 

f  x—2y  =  h  x—y=l 

3.  x-2y  =  2  4.  2a;+3y=19 

2>/-6^  =  3  3:r+2y  =  16 

5.  I2x -by =Z  6.  3:c -23^  =  3 

3ic+4y  =  3  4^+2y=5 

88.  Algebraic  solution  by  addition.  The  results  obtained 
from  the  last  examples  indicate  that  the  graphic  method, 
depending  as  it  does  upon  the  actual  measurement  of  lines, 
is  not  as  accurate  as  might  be  desired,  for  the  answers  do  not 
always  satisfy  the  conditions  exactly.  A  method  is  needed 
that  will  give  better  results. 

Consider  the  two  equations 

x-\-2y==l  (1) 

x-2y  =  b  (2) 

the  solution  of  which  was  found  to  be 
x=.Z  y=--l 

If  the  corresponding  sides  of  the  equations  be  added 
together,  the  y  disappears  and  we  have 

x+2y=^l  (1) 

x-2y  =  b  (2) 

Adding  (1)  and  (2).         2^  =  6  (3) 

x  =  Z 

We  have  cleared  equation  (3)  of  y.  We  say  we  have 
eliminated  y.  The  word  "eliminate"  is  derived  from  the 
Latin  verb  elimino,  which  means  "to  turn  out  of  doors." 

The  corresponding  value  of  y  can  then  be  found  by  the 
substitution  of  x  =  S  in  one  of  the  given  equations. 


LINEAR  EQUATIONS  IN  TWO  UNKNOWNS       127 

Substitute  it;  =  3  in  (1),  S+2y=l 

2y=-2 
y=-l 

The  work  is  checked  by  the  substitution  of  both  values 
in  the  other  equation: 

Check  by  putting  x  =  S,  y=  —I  in  (2) 


3-(2--l) 

3+2 

5 


Exercise  6  of  the  last  article  was  found  difficult  by  the 
graphic  method.     Let  us  solve  it  by  the  algebraic  method: 


dx-2y  =  S 

4:x+2y  =  5 
Eliminate  y  by  adding,        7%  =  8 

8 
^=7 
Find  value  of  y  by  substituting  in  (1), 

3 -1-2^  =  3 
7      7^ 


(1) 
(2) 
(3) 


14=^ 


(4) 


Solution  is 


^-r  y-u 


Check  by  substituting  in  (2) : 


4-?  +  2 


14 


32     3 

7"*"7 


Why  is  this  method   better  for  this  problem  than  the 
graphic  method? 


128  BEGINNERS'  ALGEBRA 

EXERCISES 

Solve  by  algebraic  method: 

1.  What  change  in  the  method  is  needed  to  eliminate  the  y  from 
the  equations? 

4a:+2y=5 
3^+23;  =  3 
Note.     Use  either  subtraction  or  addition,   according  to  which 
operation  is  needed  to  eliminate  one  of  the  unknowns. 

2.  Which  letter  would  you  eliminate  from 

Zy-2x=lZ 

3.  llw+8w  =  76  4.  7:^-3^  =  15 
lln+7m  =  72  2x-Sy  =  5 

'  5.  Sx-\-2y=ll  6.  7a-5b  =  52 

a;+23;  =  5  2a+56  =  47 

89.  More  complicated  sets.  The  equations  that  arise  in 
actual  practice  are  seldom  so  simple  as  those  of  the  last 
article,  which  have  one  term  the  same  in  both  equations. 

The  set  5t+s  =  Q  (1) 

2^-35  =  16  (2) 

is  not  of  this  common  term  type.     But  equation  (1)  can  be 

made  to  have  a  term  just  like  a  term  in  (2),  all  except  the 

sign,  if  both  sides  of  (1)  are  multiplied  by  3. 

Multiply  (1)  by  3, 

15/+35  =  18 
2/-35  =  16 
and  this  set  can  be  solved  as  before. 
Consider  a  still  more  general  case : 

2x-5y  =  l  (1) 

7x+Sy  =  24  (2) 

There  is  no  particular  choice  of  the  letter  to  eliminate. 
Let  us  choose  to  eliminate  y.  To  do  this,  we  must  make  the 
y  terms  the  same. 


LINEAR  EQUATIONS  IN  TWO  UNKNOWNS      129 

Multiply  both  sides  of  (1)  by  3, 

6x-15y  =  S 
Multiply  both  sides  of  (2)  by  5, 

S5x+15y=120 
Add,  4:lx=12S 

x  =  S 
Substitute  x  =  Sia  (1), 

Q-5y  =  l 
—5y=—5 
y  =  l 
Check. 

EXERCISES 

Solve  by  the  addition  method  and  draw  the  graphs  for  the  first 
six  exercises. 

2.  7x-{-2y=^7 

5x—4:y=l 
5.  2x-{-7y  =  SS 
3x4-43;  =  31 
8.  17x->'  =  31 

15x-\-3y=  -27 
11.  Qx-5y=13 
5x-h2y=  -20 


1. 


Sx+7y  =  27 

5x+2y  =  lQ 
4.  17^-186=15 

5a+126  =  39 
7.  5-\-p+2y  =  0 

7-\-5p-{-y  =  0 
10.  9x-2y  =  Al 

4:X-{-3y=  -9 


3.  3t;+4w  =  3 

12v-5w  =  S 
6.  2n-Zp  =  l2 

3n+5/>=-l 
9.  5x+3>'+2  =  0 

3x+2>'+l  =  0 
12.  5/+25  =  6 

4/ -35  =  37 


90.  Standard  form.  Before  attempting  to  eliminate  one 
of  the  imknowns  the  equations  should  be  put  into  the 
standard  form ,  ax-\-hy  =  c 

For  example,  4^  —  4  =  5 — 63/ 

should  be  changed  to  the  form, 

4:»;+6t  =  9 


EXERCISE    I 

Arrange  the  following  equations  in  the  standard  form: 
1.  5+2x-3  =  7y+8  2.  2x-2y-Z  =  7-x^^y 

3.  2(:r-3)H-5  =  3(y-2)  4.  7-2(:»;+3)=5-(2y-7) 

2_y 
5 


5.  Z{y-x)-\-2x  =  7 -x-2{y-^7) 


6.  1-8  = 


10 


130 


BEGINNERS'  ALGEBRA 


EXERCISE    II 

Solve: 

1.  lSx-lly  =  5-\-5x 

2.  10jc+8  =  2y 

3.  3(^-y)  =  35+3y 

4^+33;+2=l- 

-2y 

5y_44=_15:^ 

33'=15-5x 

5.-1-1 

5.  lln-y=3 

6.  270-366  =  1 
2^-7^  =  2 

7.  p-2r=8r-{-l 

8.  f-4=y 

9.  x-\-y  =  2y-7 

2p-Ar=p+r' 

-9 

3y=5(a;+l) 

91.  Algebraic  solution  by  substitution.     In  many  examples 
the  work  of  eliminating  one  of  the  unknowns  may  be  short- 
ened if  we  substitute  from  one  equation  into  the  other. 
For  instance,  a  =  36  (1) 

4a -76  =  20  (2) 

In  (1)  a  is  given  in  terms  of  b.     Substitute  this  value  36 
for  a  in  equation  (2) : 

4-36-76  =  20 
126-76  =  20 
56  =  20 
6=4 
Find  the  numerical  value  of  a  by  substituting  6  =  4  in  (1). 

a  =  3-4 
=  12 
Check  by  substituting  both  values  in  (2). 


EXERCISES 

Solve  by  substitution,  drawing  graphs  of  the  first  5: 

1.  5x-y=16  2.  3x=-2y 

x  =  y  x=S5-\-lly 

3.  n-5=-p  4.  a;=2+6y 

/>-n-l  =  3  Sy-8x  =  29 


LINEAR  -EQUATIONS  IN  TWO  UNKNOWNS      131 


5.  5x-Sy-72-. 

=  5)^ 

6. 

5x-2y  =  20 

x-l  =  15y 

x=y-2 

7.  5x-Sy  =  7 

8. 

7:^+20  =  5 -8y 

x=l+y 

x+y=  -3 

9.  6x  =  7i-3y 

10. 

7jc-6  =  5y+28 

x=  -y 

y-{-2=3x 

11.  8:x:-7y=17^ 

12. 

5^4-3y=105-3jc+10y 

.=3y-| 

^+^  =  0 

13.  7:v->'  =  21- 

2:*;- 

-5y 

14. 

4x-3y=li 

x+y=-l 

y-x=l 

15.  Ax-5y  =  '3- 

4x- 

-23' 

16. 

3y-6x  =  l 

2x  =  y-l 

3x  =  y+l 

17.  6.4y=f^ 

18. 

14:t+15>'  =  5 

Sx  =  y 

7x=5y 

19.  |+,=8 

20. 

9      2       5 
2^-f =I4 

x=5y 

2a:=9y 

21.  8^-^=1 
5 

22. 

33^-50=1 

5a:=2y 

x=2y 

o.   l+y    ^+3. 

_3 

24. 

a+2b  =  0 

^^V    4         12  - 
^=l+y 

~4 

f-.=n 

92.  Choice  of  methods.  Either  of  these  two  algebraic 
methods  of  solving  two  linear  equations  in  two  unknowns  is 
applicable  in  any  given  case,  but  there  is  often  an  advantage 
in  using  one  method  rather  than  the  other.  Use  your  judg- 
ment in  determining  which  method  you  will  use  for  any  one 
set  of  equations.  Select  the  method  that  seems  most  suit- 
able to  the  case  in  hand.    Substitution  is  the  better  method  for 

5x-y=lS 
x=2y 
Why? 


132 


BEGINNERS'  ALGEBRA 


But  for  Sx-5y=-ll 

7x—4:y  =  5 
the  addition  method  is  much  to  be' preferred.    Why?    Con- 
vince yourself  of  the  truth  of  the  suggestion  by  solving  both 
examples  by  both  methods. 


EXERCISES 

In  solving  the  following  use  the  method  that  seems  to  you  the 
most  appropriate: 


1.  40^+33;  =  6000 
:r+2y=2000 

2.  3^=6 
5^+5  =  5 

3.  24-«+2V 
32=«+^|/ 

4.  4:x;-7y  =  19 
4:X-{-9y=Q7 

5.  x-y  =  lS 
x  =  4y 

6.  y  =  3x-19 

x=Sy-7 

7.  8x-\-3y  =  ^l 
7x-5y=lS 

8.  x-i-2y=5 
2x+y=l 

9.  a  =  5b 

10.  Solve  for  x  and  y. 

5  =  26 -3a 

x+y=a 

x—y  =  b 

11.  p-2r=Sr-{-l 
2/>-4r=/>-2r+9 

12.  a+b  =  7 
Sa  =  b 

13.  2n-Sp=12 
Sn-5p  =  17 

14.  — y  =  5 

x-'y  =  7 

15.  2x-y=l 
7x+2y=5 

16.  3»-2w  =  21 
n    fjf    5 
4     5~4 

17.  7x-{-^y  =  3 

18.  .la;-.01y=.296 

x-^y=l 

af+>'=10 

93.  Problems  solved  by  means  of  two  unknowns.    The 

sum  of  two  numbers  is  33,  the   difference  is   17.    What 


LINEAR  EQUATIONS  IN  TWO  UNKNOWNS       133 

are  the  numbers?     This  problem  may  be  solved  in  two 

ways. 

(a)  With  one  unknown: 

The  equality  is 

the  larger — the  smaller  =  17 
Let  the  smaller  nvimber  =  :3f,  then  the  larger  =  33— a? 
Hence  the  equation    3S—x—x  =  17 
33-2;t  =  17 
16  =  2it; 

S  =  x,  the  smaller 
Hence  33-8  =  25,  the  larger 

(6)  With  two  unknowns : 

The  problem  fiunishes  two  equalities : 

the  larger + the  smaller  =  33 

the  larger — the  smaller  =  17 

Using  X  for  the  larger  number  and  y  for  the  smaller  num- 
ber, we  have  x-\-y  =  S3  (1) 

;t:->'  =  17  (2) 

Eliminate  y  by  addition,      2x  =  50 

x  =  25,  the  larger 

Substitute  ic  =  25  in  equation  (1), 
25+>;  =  33 

y  =  S,  the  smaller 

Which  method  do  you  regard  as  the  better  in  this  case? 
Why? 

Many  problems  can  be  solved  in  both  ways,  and  often 
there  is  little  choice  between  the  two  methods.  In  many 
cases,  however,  the  method  of  two  unknowns  is  to  be  pre- 
ferred; in  some  cases  it  alone  can  be  used  successfully.  To 
be  solved  by  two  unknowns  the  problem  must  contain  two 
separate  statements  concerning  the  two  unknowns. 


134  BEGINNERS'  ALGEBRA 

PROBLEMS 

Number  problems: 

1.  The  sum  of  two  numbers  is  33  and  their  difference  is  7.  What 
are  the  numbers?     Draw  graphs. 

2.  The  sum  of  two  numbers  is  32  and  their  difference  is  47. 
What  are  the  numbers? 

3.  The  sum  of  two  numbers  is  35  and  their  difference  is  20. 
What  are  the  numbers? 

4.  The  sum  of  two  nimibers  is  m  and  their  difference  is  n.  What 
are  the  numbers? 

5.  Apply  the  formula  found  in  the  last  problem  to  the  special 
cases:  the  sum  is  18,  difference  is  12;  sum  is  16,  difference  is  20; 
sum  is  97,  difference  is  63;  sum  is  3675,  difference  is  2691.  The 
formula  is  a  rule  for  finding  two  numbers  when  their  sum  and  their 
difference  are  given.     State  the  rule  in  words. 

6.  The  sum  of  two  integers  is  87  and  their  difference  is  32. 
What  are  the  integers? 

7.  The  sum  of  two  numbers  is  27;  twice  the  first  added  to  3  times 
the  second  is  39.    What  are  the  numbers? 

8.  The  difference  between  two  numbers  is  28;  5  times  the  first 
less  the  second  is  197.    What  are  the  numbers? 

9.  The  sum  of  two  numbers  is  twice  their  difference,  and  twice 
the  larger  one  is  7  more  than  5  times  the  smaller  one.  What  are 
the  nimibers?    Draw  graph. 

Digit  problems: 

The  digits  or  figures  of  93  are  the  same  as  those  of  39, 
but  they  are  reversed  in  order.  If  x  represents  the  tens 
digit  and  y  represents  the  units  digit  of  a  number,  what 
represents  the  nimiber  itself?  What  will  represent  the  num- 
ber with  the  digits  reversed? 

10.  The  sum  of  the  digits  of  a  number  is  10.  If  36  is  subtracted 
from  the  number,  the  digits  will  be  reversed.     Find  the  number. 

Solution: 
Each  of  the  first  two  sentences  of  the  problem  states  an  equality. 

(1)  One  digit+other  digit  =  10 

(2)  The  number  -  36  .=  a  number  with  the  same  digits  but  reversed 


LINEAR  EQUATIONS  IN  TWO  UNKNOWNS      135 

The  two  digits  are  the  unknowns. 

Let  the  tens  digit  =  x  and  the  units  digit  =  y. 

Translate  into  the  equations 

(1)  x+y=\0 

(2)  lOx-\-y-m=lQy-{-x 
and  solve. 

11.  The  tens  digit  of  a  certain  number  is  2  less  than  twice  the 
units  digit.  If  27  is  subtracted  from  the  number,  the  digits  will  be 
reversed.    Find  the  number. 

12.  The  tens  digit  of  a  certain  number  is  1  more  than  the  units 
digit.  The  number  itself  is  6  times  the  sum  of  the  digits.  Find 
the  number. 

13.  The  tens  digit  of  a  certain  number  is  twice  the  imits  digit. 
The  number  itself  is  6  less  than  12  times  the  tens  digit.  Find  the 
number. 

14.  The  tens  digit  of  a  certain  number  is  1  less  than  twice  the 
units  digit.  If  18  is  subtracted  from  the  number,  the  digits  are 
reversed.    Find  the  number. 

15.  The  units  digit  of  a  certain  number  is  2  less  than  the  tens 
digit.  The  number  formed  by  reversing  the  digits  is  7  times  the 
tens  digit.    Find  the  number. 

16.  The  tens  digit  of  a  certain  number  is  2  more  than  the  imits 
digit.  The  nimiber  formed  by  reversing  the  digits  is  3  less  than 
12  times  the  units  digit.     Find  the  number. 

17.  The  tens  digit  of  a  certain  number  is  4  less  than  twice  the 
units  digit.  The  number  itself  is  2  more  than  6  times  the  sum  of  the 
digits.    Find  the  number. 

18.  The  simi  of  the  digits  of  a  nimiber  is  s.  When  n  is  added  to 
the  number,  the  digits  are  reversed.  Find  the  following  formulas 
for  computing  the  digits  of  the  number,  x  being  the  tens  digit  and 
y  being  the  units  digit. 

*~   18  ^~    18 

Mixture  problems: 

19.  How  many  pounds  each  of  nuts  at  20  cents  a  poimd  and  nuts 


136  BEGINNERS'  ALGEBRA 

at  45  cents  a  poimd  should  be  mixed  to  make  up  10  pounds  worth 
35  cents  a  pound? 

Suggestion.  The  two  equalities  are  not  clearly  evident,  but  they 
can  be  found  by  a  little  careful  thinking ;  one  will  be  between  numbers 
of  pounds  and  the  other  between  values.     Evidently, 

no.  of  lbs.  of  first  kind+no.  of  lbs.  of  second  kind  =  10 
cost  of  first  kind  +cost  of  second  kind  =  cost  of  mixtiu"e 
Choose  as  the  unknowns  for  the  equations  the  number  of  pounds  of 
each  kind  used.     Translate  equalities  and  solve. 

20.  How  many  pounds  each  of  40-cent  coffee  and  60-cent  coffee 
must  be  mixed  to  make  12  pounds  of  45-cent  coffee? 

21.  How  much  candy  worth  45  cents  a  pound  and  candy  worth 
85  cents  a  pound  must  be  mixed  to  make  20  pounds  worth  60  cents 
a  pound? 

22.  If  a  and  b  are  the  costs  per  pound  of  two  articles  that  are  to 
be  mixed  to  make  a  mixture  worth  c  a  pound,  find  formulas  for 
calculating  the  quantity  of  each  to  make  a  mixture  of  n  pounds. 


CHAPTER  VII 

Special  Products;  Factoring;  Equations 
Solved  by  Factoring 

94.  A  problem.  In  the  foregoing  chapters  we  have  seen 
that  certain  problems  can  be  solved  by  means  of  one 
equation  in  one  unknown,  while  others  may  be  solved  by 
two  Hnear  equations  in  two  unknowns.  There  are  many 
problems  that  lead  to  other  kinds  of  equations  more  or  less 
complicated.  As  an  illustration  consider  the  following 
problem : 

A  man  wishes  to  double  the  area  of  an  8  by  12  rod 
field  by  adding  to  one  end  and  one  side  strips  of  equal 
width.  How  wide  must  these 
strips  be? 

If  we  let  s  represent  the  width 
of  strip  to  be  added,  the  plan  of 
addition  is  given  in  Fig.  33. 
The  area  of  the  added  part 
equals  the  area  of  the  original 
field.  Fig.  33 

That  is  125+5  •  5+85=  12  •  8 

or  55+205  =  96 

This  equation  is  very  different  from  any  we  have  con- 
sidered thus  far,  for  the  unknown  appears  twice  in  one  term. 
Any  attempt  you  may  make  to  solve  it  will  soon  lead  you 
to  see  that  none  of  the  methods  of  solving  equations  you 
have  yet  learned  wiU  apply  here.  We  need  an  entirely 
new  method  for  the  solution  of  this  equation.  Before 
we  can  attack  the  problem  of  finding  a  new  method  with  any 
hope  of  success,  it  will  be  necessary  for  us  to  develop  a  little 
more  algebraic  machinerv  involving  both  new  ideas  and  new 
ways  of  working. 

10  137 


S                12-5           5 

5  .5 

12 
12-8        8 

5 
8.5 

5 

138  BEGINNERS*  ALGEBRA 

95.  Powers.  Exponents.  In  the  product  2X3  =  6,  2  and 
3  are  factors  of  6.  Each  of  two  or  more  numbers  whose 
product  is  a  given  number  is  called  a  factor  of  that  number. 

A  number  that  is  the  product  of  two  or  more  equal  factors 
is  called  a  power  of  one  of  the  equal  factors. 

25  =  5  •  5  is  the  second  power  of  5. 

8  =  2  •  2  •  2  is  the  third  power  of  2. 

243  =  3  •  3  •  3  •  3  •  3  is  the  fifth  power  of  3. 

bb  is  the  second  power  of  b. 

aaaa  is  the  fourth  power  of  a. 

In  the  latter  part  of  the  sixteenth  century  Simon  Stevin 
introduced  a  better  way  of  writing  a  power.  In  place  of  a 
string  of  letters  or  numbers,  one  factor  is  used  with  a  number 
written  to  the  right  and  above  the  factor  to  indicate  how 
many  equal  factors  there  are. 

5-5  =  52  2-2-2  =  23 

bb  =  b^  aaaa  =  a* 

The  nimiber  indicating  the  number  of  factors  is  called  an 
exponent,  b^  is  read  "the  second  power  of  6"  or  "b  second 
power "  or  more  commonly  "the  square  of  " 6 "  or  " 6  square." 
a^  is  read  "the  third  power  of  a "  or  " a  third  power "  or  more 
commonly  "a  cube."  a*  is  read  "the  fourth  power  of  a" 
or  "a  foiuth  power." 

Those  who  are  of  an  inquiring  turn  of  mind  might  ask 
why  the  special  names  "a  square,"  "a  cube"  are  used  for 
a^  and  a^.     Can  you  answer  the  question? 

EXERCISES 

1.  Find  values  of  2^  5\  7\  2^  25^. 

2.  Write  3^  in  two  other  ways. 

3.  Write  2^  in  two  other  ways. 

4.  Write  4  X4  X4  in  two  other  ways. 

5.  Write  a^  in  two  other  ways. 

6.  Write  nnnnnnn  in  another  way. 


PRODUCTS;  FACTORING;  EQUATIONS  139 

7.  If  a  is  7,  what  does  a^  equal? 

8.  If  n  is  3,  what  does  w'  equal? 

9.  If  a  is  f ,  what  does  a^  equal? 

10.  If  0:  =  ^,  what  does  x^  equal? 

11.  11  x=  —2,  what  does  x"^  equal? 

12.  If  ic=  —  2,  what  does  x^  equal? 

13.  If  ic=2.5,  what  does  x"^  equal? 

14.  If  a=  .05,  what  does  a^  equal? 

15.  If  0  =  3  and  h  =  2,  find  value  of  a2+62,  a^-¥,  a-h'^-^-a^, 

16.  Find  value  of  3  X5^  3  •  2^  •  5''. 

17.  Find  value  of  5o'  when  a  =  2. 

Note.  It  must  be  carefully  noted  that  in  such  an  expression  as 
5a3  the  exponent  affects  only  the  letter  over  which  it  is  placed. 

5a'  means  5  -  a  -  a-  a 
Ifa  =  2,  5a3means5- 2- 2- 2 

18.  Find  value  of  2a^-{-Sa  when  a  =  5. 

19.  Find  value  of  120:^^  when  x=  —2,  y  =  ^. 

20.  Find  value  of  2ax^—3x-\-5  when  x  =  5,  a  —  1. 

21.  Draw  the  graphs  of  x^,  x^,  x-\-2,  x. 

96.  Names  of  algebraic  expressions.  An  algebraic  expres- 
sion having  but  one  term  is  called  a  monomial.  An  algebraic 
expression  having  two  terms  is  called  a  binomial.  One 
with  three  terms  is  called  a  trinomial. 

3^2  is  a  monomial. 
2a!:4-3  is  a  binomial. 
2x^—Zx-\-2  is  a  trinomial. 

97.  Products  of  monomials.  The  product  of  2a  and  Zx 
is  2a  •  Z%  or,  better,  ^ax.  In  like  manner  the  product  of 
2ax'^  and  ?tax  is 

2a^^  •  3a::i;  =  2'  a'  X'  X'Z*  a*  x 
or,  rearranging,  =2*Z'  a-  a'  X'  x*  x 

or,  using  better  notation,       =  Wx? 


140  BEGINNERS'  ALGEBRA 

It  is  unnecessary  to  write  out  all  the  factors  as  is  done 
above.  All  that  is  necessary  is  to  count  up  the  number  of 
times  each  letter  occurs  as  a  factor.  This  can  be  done  in 
the  simplest  way  by  the  addition  of  the  exponents  of  that 
letter,  thus: 

=  lOaH^ 

This  way  of  finding  the  product  of  two  monomials  may  be 
stated  as  a  rule : 

Rule.  The  product  of  two  monomials  is  a  monomial 
in  which  the  numerical  coefficient  is  the  product  of  the 
numerical  coeflBlcients  of  the  factors  and  in  which  the  expo- 
nent of  each  letter  is  the  sum  of  the  exponents  of  that  letter 
in  the  factors. 

If  a  letter  has  no  exponent  placed  over  it,  it  is  to  be  under- 
stood that  the  exponent  is  one.  This  is  evident  from  the 
purpose  for  which  an  exponent  is  used. 

EXERCISES 

3.  2x  •  3x2 
6.   -X  •  -2x» 
9.  bat  •  3^2/ 
12.  6ax  •  5a2 
15.   -5aw2  •  2a^n 
18.  5an  •  -2a» 
21.  (3xy2)2 

Evaluate  the  following  when  a  =  3,  n=  —3,  x=  —2: 

22.  3a2w,   San\   Sanx,   2a*n,   2ahi\   6n^x,  6nx^,   5aw',   5ahi 

23.  2x2  •  3a,   3^2  •  2x,   4x  •  2w2,    2nx  •  x2,   5ax  •  w,   5ahi  •  n^ 

98.  Quotient  of  monomials.  If  one  factor  of  a  product  is 
given,  the  other  may  be  found  by  the  division  of  the  product 
by  the  given  factor. 

For  examole,  suppose  we  have, 

7^2.  (?)=  35:^3 

or  35^4-7a;2=? 


Find  the  products: 

1.  2x  •  5x 

2. 

3a  •  7ax. 

4.   -3«-8«3 

5. 

x2  •  ax 

7.  7an  •  2a^n 

8. 

— 3a>'  •  7a-y 

10.  2ax  •  3o 

11. 

8x2 .  _^x 

13.  4a2x  •  2x 

14. 

3x2 .  _2x' 

16.  -4a2 .  -5n2 

17. 

3a2 .  -6aw2 

19.  3w»  •  3^2 

20. 

2ax3  •  2ax3 

PRODUCTS;  FACTORING;  EQUATIONS  141 

As  division  is  the  reverse  of  multiplication,  we  have  simply 
to  reverse  the  rule  for  multiplication,  dividing  coefficients 
instead  of  multiplying  and  subtracting  exponents  instead  of 
adding: 

so  also,  12;c5  -^  Sx^  =  yr^^  =,  4^ 

This  may  be  stated  as  a  rule: 

Rule.  To  divide  one  monomial  by  another,  divide  the 
numerical  coefficient  of  the  dividend  by  the  numerical 
coefficient  of  the  divisor  and  subtract  the  exponent  of  each 
letter  in  the  divisor  from  the  exponent  of  that  letter  in  the 
dividend. 

As  we  have  thus  far  used  only  positive  numbers  for  expo- 
nents, this  rule  applies,  at  present,  only  to  cases  when  the 
exponent  of  the  letter  in  the  divisor  is  less  than  that  of  the 
same  letter  in  the  dividend.  If  the  exponents  are  the  same 
and  the  resulting  exponents  become  zero,  the  meaning  is 
simply  that  that  letter  does  not  appear  in  the  result. 

EXERCISES 


1.  If  a  product  is  35  and  one  factor  is  7,  what  is  the  other  factor? 

2.  If  3x  is  one  factor  of  6^2,  what  is  the  other  factor? 

Divide: 

3.  10x3  by  5^2 

4.  24x*  by  4:X 

5.  35x*  by  7x» 

6.  Sax^  by  ax 

7.  27ab^x  by  9ab 

8.  ^5nx*  by  9nx 

9.  -28ny^hy7y'- 

10.   -9«/5  by  -3/ 

11.  5103/6  by  3^ 

12.  63/t2/3  by  -9/// 

13.  lOjc^yby  -2x 

14.  lOx^y  by  -2xy 

15.  10x'-yhy5x'' 

16.  25a262  by  -5ab 

17.  25a^b^  by  -5a^b 

18.  lQax*hy8ax'- 

19.  18a^x*hy9ax^ 

20.  S2a^b^  by  Aab 

21.   -48x3by-8x2 

22..-51nA»by  -3h 

23.   -72aw;Kby  36aw 

24.  Evaluate  the  following  if  x=  -2,  a  = 

=  3,  b=-3: 

I5ax^      18a2 

X      IQa^b^      Ua^bx 

\2ab^      20ab^ 

ax  '       ax  '      4o6  '       Tab  '       36   '      46^ 


142  BEGINNERS'  ALGEBRA 

99.  Factoring  numbers.  When  one  of  the  factors  of  a 
product  is  given,  the  other  factor  is  found  by  division.  The 
process  of  finding  the  factors  of  a  product  when  only  the 
product  is  given  is  called  factoring. 

You  are  more  or  less  familiar  with  factoring  in  arithmetic. 
You  know  that  the  factors  of  35  are  5  and  7  because  you 
remember  from  the  multiplication  table  that  5X7  =  35. 

EXERCISES 

1.  Find  two  factors  of  56,  72,  81,  121,  32,  54,  84,  96. 

It  is  a  little  more  difficult  to  factor  numbers  that  do  not  appear 
in  the  multiplication  tables  we  have  learned.  In  such  cases  one 
has  to  experiment  with  different  numbers  as  possible  factors. 

2.  What  are  the  factors  of  39,  91,  51,  65,  221,  323? 

There  are  simple  tests  for  determining  whether  a  number  is 
divisible  by  some  of  the  smaller  integers  which  often  help  in  factor- 
ing numbers.  An  even  number  is  divisible  by  2.  Why?  A  num- 
ber ending  with  0  or  5  is  divisible  by  5.  Why?  If  the  sum  of  the 
digits  of  a  number  is  divisible  by  3,  the  number  is  divisible  by  3. 
If  the  simi  of  the  digits  of  a  number  is  divisible  by  9,  the  number  is 
divisible  by  9. 

3.  Factor  62,  115,  111,  171,  207,  57,  306,  855,  269,  441,  385, 
495,  234,  104,  408,  195. 

100.  Prime  factors.  A  prime  number  is  an  integer  that  is 
exactly  divisible  by  no  integer  except  itself  and  one. 

A  number  is  said  to  be  factored  into  its  prime  factors 
when  all  the  factors  whose  product  is  the  given  number  are 
prime.     Thus: 

28  =  7-2-2 

Exponents  may  be  used  for  repeated  factors: 
28  =  7-22 

Factor  into  prime  factors  18,  36,  98,  75,  250,  240,  512, 
288,  350,  360,  448,  108,  196,  225,  175,  484,  728,  357,  372,  450. 


PRODUCTS;   FACTORING;  EQUATIONS  143 

101.  Factors  of  monomials.  6a62  is  a  monomial.  Such  an 
expression  is  already  in  a  factored  form;  it  is  the  product 
of  several  factors.  These  factors  can  be  separated  in  various 
ways: 

Into  two  factors  such  as  6a  •  6^  or  a  •  Qb^ 
Into  three  factors  such  as  6  •  a  •  6^ 
Into  prime  factors  such  as  2  •  3  •  a  •  6  •  6 

EXERCISES 

1.  Separate  into  prime  factors  lOax,  15bx'^,  S2ax'^y  12aH^,  2oabx^. 

2.  Separate  into  two  factors  42,  7xy,  15a\  90,  ISax^,  S6aH. 

3.  Separate  24ax2  into  two  factors  one  of  which  is  3  a. 

4.  Separate  24iax^  into  two  factors,  one  being  6ax. 

5.  Separate  S5x^  into  two  factors  one  of  which  is  7a:*. 

6.  Separate  bla'^n^  into  two  factors  one  of  which  is  San. 

7.  Find  a  number  that  is  a  factor  of  both  Qx  and  3a. 

8.  Find  a  nimiber  that  is  a  factor  of  both  ax  and  bx. 

9.  Find  a  factor  that  is  common  to  both  Ax  and  Qx^.  How  many 
can  you  find? 

102.  Product  of  a  binomial  and  a  monomial.  We  have 
seen  that 

3(:x;-5)=3:c-15 

a(x-\rb)  =ax-\-ab 
x{x+2)=x^-\-2x 
In  each  case  there  are  two  forms  for  the  same  number. 
To  find  the  second  form,  multiply  each  term  of  the  binomial 
by  the  monomial. 

The  same  method  may  be  extended  to  the  multipHcation 
of  an  expression  of  any  number  of  terms  by  a  monomial: 

S{x^+x-2)=Sx''-\-Zx-Q 

EXERCISES 

Multiply: 
1.  S(x-2)  2.  :^(:»;-3)  3.  2x(x-5) 

4.  -3:x;(2:r-3)  5.  w(n-a)  6.  »(n»+2) 


144  BEGINNERS'  ALGEBRA 

7.  ^a{Z-b)  8.  2a{a+b)  9.  a{a^-5) 

10.  a\a-b)  11.  xix^-x+2)  12.  a;2(2:x:+5) 

13.  2;c2(5x-2)  14.  3w(2+w3)  15.   _5a(2a-36) 

16.  m{2m-n)  17.  a(6-c)  18.  a2(a+6) 

19.  -Jt:2(2»3-x+5)  20.  ^aH^x'' -bx-\-2>)  21.   -3x(;x:2_i) 

103.  Eauivalent  forms.     The  expressions 

a(^+6),        ax+ab 

are  two  forms  for  the  same  ntimber: 

a{x-{-b),  SL  product  form  showing  two  factors 
ax-\-ab,  a  sum  form  showing  two  terms. 

We  have  akeady  found  it  useful  to  change  a  product  form 
such  as  S{x+4:)  into  the  form  3^+12  in  order  to  solve  the 
equation  in  which  the  expression  appeared.  We  shall  find 
the  reverse  process  of  undoing  the  multiplication  just  as 
useful.  This  reverse  process  is  called  factoring.  It  con- 
sists in  changing  a  sum  of  several  terms  into  the  product  of 
several  factors;  that  is,  a  sum  form  into  a  product  form. 

It  should  be  noticed  that  the  factoring  of  Art.  99  is  also 
changing  a  sum  form  into  a  product  form. 

Factor  35: 

35  means  30+5,  a  sum 

5  •  7  is  a  product  form 

If  one  of  the  factors  is  known,  the  other  factor  is  easily 

found  by  division  as  is  shown  in  the  next  article. 

104.  Division  by  a  monomial.  If  one  factor  of  7;c— 21  is 
7,  the  other  is  found  by  division  of  each  term  of  7^  — 21  by  7. 
It  is  x-S. 

Hence  7::i;-21  =  7(:;f-3) 

EXERCISES 

Divide: 
1.  27a  -9  by  9  2.  Sax -a  by  a 

3.  7o-14&by-7  4.  a^+a  by  a 


PRODUCTS;  FACTORING;  EQUATIONS  145 

5.  8a2 -4a  by  4o  6.  da+Sa^by-a 

7.  15a^-10ax  by  5a  8.  7a+Uay  by  7a 

9.  15x2-25:c+10by-o  10.  Sx^a+^^b-^-x^hy  x^ 

11.  Ux^a -21x^a^+2Sxa^  by  7 ax 

12.  oaj2 - bx^ -ex*  by- x^ 

13.  If  one  factor  of  12a^x—4ibx  is  4a!;,  what  is  the  other  factor? 

14.  One  factor  of  9a:c+12a2jc  is  3ax.     Find  the  other  factor. 

15.  One  factor  of  9a*-27a^-lSa  is  9a.     Find  the  other. 

105.  Factoring.  The  question  of  changing  a  sum  form 
into  a  factored  form  when  no  one  of  the  factors  is  given  is  a 
much  more  difficult  matter.  All  expressions  cannot  be  so 
changed.  Only  certain  kinds  can  be  changed  in  this  way, 
or,  as  we  commonly  say,  can  be  factored.  We  shall  con- 
sider in  this  book  only  a  few  of  the  simpler  kinds  or  types. 
We  discover  such  types  when  we  compare  the  result  of  a 
multiplication  with  the  factors  from  which  it  came. 

106.  Common  factor  tjrpe.    a(6+3)=a6+3a 

The  factor  a  is  in  every  term  of  a6+3a.  It  may  then  be 
chosen  as  one  factor  and  the  other  factor  found  by  the 
division  of  a6 + 3a  by  a .  So  in  ic^ — 4jt; ,  ic  is  a  nimiber  common 
to  the  two  terms. 

Hence  x^  —  4:X  =  x(x  —  ^) 

So  3ax+12a  =  Sa{x-\-^) 

Any  expression  in  which  the  terms  do  not  have  a  common 
letter  or  number  cannot  be  factored  in  this  way. 

EXERCISE  I 

Factor  and  then  check  the  results  by  multiplying : 

1.  5«+35       ^  2.  5a+156  3.  a^+3a 

4.  5a^-18a  5.  St+P  6.  57r-25 

7.  7;c-14  8.  2:r-3  9.  Ta^-irb'' 

10.  Sn^+21  11.  7w2-14«  12.  2a -4b 

13.  a^TF-a  14.  12:r-8a:«  15.  3n^-2n 


146 


BEGINNERS'  ALGEBRA 


16.  25+65 

19.  7w-6 

22.  ax-^bx 

25.  3w»-15w2-|-6w 

28.  ax^-\-bxy 

31.  25 -37+25 -19 

34.  ax^-\-ax 

37.  ax^-bx^ 

40.  1052w+5w»2 


17.  72+27 
20.  4w3-12w2 
23.  7:i;2^51a; 
26.  2»2-4«-3 
29.  5w2-25w 
32.  36- 7+18- 5 
35.  abc+ab 
38.  47ra2-7r6» 
41.  8:^3 -4:^2 _^ 


18.  257r+20 
21.  2a+4ac 
24.  3«2-9w-6 
27.  Sm^-Qmn 
30.  TTab—wa^ 
33.  12 -21 -28 -5 
36.  ax-\-bx—cx 
39.  7+35^  + 161^1  • 
42.  10x^y  —  5xy 


EXERCISE  II 

1.  Draw  the  graph  of  2:r— 3. 

2.  Draw  the  graph  of  oc  —  4. 

3.  Draw  the  graph  of  x^  -4x.     (See  Fig.  34.) 

Ux=     6    1    5    I    4    I    3    I    2    I    1    I    0 


Then:x:2-4^= 


L2 

5 

0 

- 

-3 

-4| 

-t 

5 

0 

' 

^ 

, 

A 

^ 

^ 

" 

3 

' 

■' 

12 


Fig.  34 

The  points  on  the  figure  are  rather  far  apart.     Let  us  find  other 
points  in  between  them.     (See  Fig.  35.) 

Take  x  = 
Then  x^-'ix^ 


5i  1  H     3i     2i     li      i      -i  -li 

8il  2i   -l|-3ii-3il    ?       ?       ? 

PRODUCTS;  FACTORING;  EQUATIONS 


147 


You  may  insert  more  points  if  you  desire,  but  these  are  sufficient 
to  show  the  form  of  the  smooth  curve  that  can  be  drawn  through 
these  points.     (Fig.  36.) 


~ 

"~ 

~ 

"~ 

— 

— 

— 

■~ 

— 

~" 

~ 

~ 

— 

— 

■ 

"■ 

i 

>  i-< 

~ 

~ 

— 

'~ 

1 

"' 

' 

"■ 

" 

1 

n 

~ 

~ 

I 

1 

"■ 

'f 

I" 

n 

n! 

' 

P 

o 

n 

1 

I 

X 

uL 

^    1 

0 

■ 

~ 

'" 

■' 

0 

T" 

~ 

" 

n 

n 

b 

\ 

L 

0 

f) 

k 

~ 

~ 

^ 

■ 

" 

■■ 

■ 

u 

" 

" 

' 

"■ 

' 

_ 

_ 

Ij 

1 

_ 

1 

_] 

- 

Fig.  35 


Fig.  36 


You  will  notice  that  the  graphs  of  Exercises  2  and  3  arc 
quite  different  in  form. 

Expressions  of  the  kind  x—4:  axe  said  to  be  of  the  first 
degree.    Why? 

Expressions  of  the  kind  x^—ix  are  said  to  be  of  the  second 
degree.  Why?  How  do  you  pick  out  an  expression  of 
the  second  degree? 

What  is  the  degree  of  x^-Sxf    Why? 

Expressions  of  the  first  degree  are  often  called  linear 
expressions.  An  expression  of  the  second  degree  is  com- 
monly called  a  quadratic  expression. 

After  you  have  drawn  the  graphs  of  the  following  expres- 
sions what  conclusion  do  you  think  you  could  safely  make? 

4.  x*-6x  5.  a;2+4a;  6.  2a:«-5jc 

7.  ix^+6x  8.  xix-3)  9.  x{x-[-5) 


148  BEGINNERS'  ALGEBRA 

107.  Checking  work  by  substitution  in  identities.    The 

statement 

«(m--5)=w2— 5n 
is  an  identity.     The  right  side  is  but  another  form  for  the 
number  represented  by  the  left  side. 

If  we  let  n  =  7 

we  have 


7(7-5) 

72-5-7 

7-2 

49-35 

14 

14 

If  n  =  10,  what  is  the  result? 

If  w  =  2,  what  is  the  result? 

What  conclusion  do  you  think  it  is  safe  to  make?  It 
should  be  noted  that  the  numerical  calculation  of  the  two 
sides  must  be  worked  but  by  different  methods.     Why? 

To  check,  put  x  =  d,  a  =  2 


3+6  •2-32 

9+108 

117 


3  •  3(1+2  •2-3) 

9-  13 

117 


Try  checking  with  some  other  values  for  x  and  a. 

Thus  we  may  check  the  truth  of  any  identity  by  sub- 
stituting some  ntimber  for  each  letter  and  working  out  the 
two  sides  independently.  If  the  two  sides  come  out  to  be 
the  same  number,  it  is  safe  to  say  that  the  identity  is  true , 
provided,  of  course,  that  you  have  made  no  mistakes  in  the 
numerical  calculations. 

EXERCISES 

In  the  following  exercises  multiply  or  factor  as  the  case  may  be 
and  check  your  results: 

1.  3(x-2)  2.  2a{Sx+a)  3.  2ic2(«-2) 

4.  6x+Sy  5.  6x-Sx^  6.  2a(Sa-Qb) 

7.  x(2x'^-x-\-3)  8.  3x>-Qx^  9.  3x^-9x 


PRODUCTS;  FACTORING;  EQUATIONS  149 

If  in  the  following  exercises  any  are  untrue,  write  in  correct  form 
and  check: 

10.  x(x^-Sx)  =  x^-Sx  11.  2a{a-h)  =  2a^-2ab 

12.  2a{x-a)  =  2ax-2a  13.  3a;2+6a  =  3:c(a;+2a) 

14.  \bx''-l2x  =  bx{Zx-A)         15.  7jc+21a=7(jc+3a) 
16.  8x-4a:2=4jc(2+:^)  17.  6:c-2  =  2(2x-l) 

108.  A  factor  equal  to  zero.  In  drawing  the  graph  of 
such  products  as  x{x  —  ?t)  it  will  be  noticed  that  when  one  of 
the  factors  is  zero  the  product  will  be  zero. 

This  is  an  instance  of  a  general  principle  of  very  great 
importance,  namely,  a  product  is  zero  when  and  only  when 
one  of  its  factors  is  zero. 

If  a  factor  is  zero,  the  product  must  be  zero ;  if  a  product 
is  zero,  one  or  more  of  its  factors  must  be  zero. 

EXERCISES 

1.  If  X  is  7,  what  is  the  value  of  xix-7)\  3(:»-7)? 

2.  If  X  is  0,  what  is  the  value  of  x{x  -3) ;  7(^  -3)? 

3.  For  what  values  of  x  are  the  following  products  zero? 

3(^-2),  4(:i:+3),  x{x-l),  x{x-7),  x{x-\-2),  {x-S)x 

109.  A  problem  solved.  We  are  now  prepared  to  solve  a 
problem  leading  to  an  equation  of  the  kind  suggested  at 
the  beginning  of  this  chapter.  Consider  a  problem  like  this : 
The  difference  between  two  numbers  is  7 ;  twice  the  square 
of  the  smaller  equals  their  product.     What  are  the  numbers  ? 

The  equality  is 

twice  (smaller)2  =  smaller  X  larger 
Let  X  =  smaller  number 
then  x+7  =  larger 

and  the  equality  becomes 

2x^  =  x(x-\-7) 
.    Expanding,  2x^-^x'^^-7x 

Collecting  terms  on  one  side, 

2x^-x^-7x  =  ^ 
or  x^-7x  =  ^ 


150  BEGINNERS'  ALGEBRA 

This  equation,  rr^  — 7^  =  0,  asks  the  question:  For  what 
values  of  x  is  x^  —  lx  equal  to  zero?  The  answer  is  to  be 
found  by  the  method  of  putting  x^  —  1x  in  the  factored  or 
product  form  and  applying  the  principle  given  in  the  last 
article : 

Thus,  x^-lx  =  {) 

Factoring,  x(x—l)=0 

The  product  is  zero  if  either  factor  is  zero ;  that  is, 

ifr^  =  0 
orif:r-7  =  0 
that  is,  x  =  7 
Both  x  =  {)  and  x  =  l  satisfy  the  equation,  as  shown  below: 
lf:x;  =  0  lix  =  7 


02 

0(0+7) 

2-72 

7(7+7) 

0 

0 

2-49 

7-14 

98 

98 

There  are  then  two  different  answers  to  the  problem  itself. 
There  are  two  sets  of  numbers  that  fulfil  the  requirements 
of  the  problem,  namely,  0  and  7,  or  7  and  14. 

110.  Root  of  an  equation.  Any  number  that  will  reduce 
an  equation  to  an  identity  when  that  number  is  put  in 
place  of  the  unknown  is  called  a  root  of  the  equation. 
An  equation  of  the  first  degree  in  one  imknown  has  but 
one  root.  An  equation  of  the  second  degree  in  one  unknown 
has  two  roots. 

In  the  solution  of  an  equation  of  the  second  degree  both 
roots  should  be  found,  as  one  is  just  as  important  as  the 
other. 

EXERCISES 

Solve  and  check: 

1.  x{x-2)  =  0  2.  x{x^-Z)  =  0 

3.  2x{x-b)=0  4.  2x{Zx-7)  =  0 

5.  fj2+9^j  =  o  6.  3:c2+9:v  =  0 

7.  96-2  =  45  8.  10n^  =  7n 

9.  0  =  3r-6r2  10.  7a;-3  =  21a;«-3 


PRODUCTS;  FACTORING;  EQUATIONS  151 

11.  S-2x'  =  5-{-5x-2  12.  8:^2 _9  =  2x -3(6^+3) 

13.  12-ll:i;2  =  8-2(ll:x;-2)     14.  ^'+1  =  2-^^ 

15.  Sx^-\-x-7  =  8x+x^-7 

16.  The  product  of  a  certain  number  and  3  less  than  its  double 
is  7  times  the  number.     What  is  the  number? 

17.  The  product  of  a  certain  number  and  itself  increased  by  5 
is  twice  the  square  of  the  number.     What  is  the  number? 

18.  Divide  10  into  2  parts  so  that  4  times  the  product  of  the  2 
parts  shall  equal  the  square  of  the  first  part. 

19.  Find  the  side  of  a  square  such  that  the  area  of  the  square 
shall  be  numerically  equal  to  twice  its  perimeter. 

20.  The  area  of  a  certain  square  is  numerically  equal  to  f  of  its 
perimeter.     Find  the  side  of  the  square. 

21.  The  width  of  a  certain  rectangle  is  f  of  its  length.  The 
area  is  numerically  equal  to  6  times  its  perimeter.  Find  its  dimen- 
sions. 

22.  If  4  is  subtracted  from  9  times  a  number  and  this  difference 
is  divided  by  4,  the  result  is  one  less  than  9  times  the  square  of  the 
number.     Find  the  number. 

23.  The  volume  of  a  certain  cube  is  numerically  equal  to  -§■  of 
its  surface.     Find  an  edge  of  the  cube. 

24.  The  volume  of  a  certain  cube  is  numerically  equal  to  a 
times  its  surface.    Find  the  edge  of  the  cube. 

25.  What  is  the  answer  to  Exercise  24  if  a  is  8,  5,  3? 

26.  The  imits  digit  of  a  certain  number  is  twice  the  tens  digit. 
The  number  itself  is  3  times  the  square  of  the  tens  digit.  Find  the 
number. 

27.  x^-cx  =  0  28.  x^=ax 
29.  bx^  =  x  30.  ax^  =  bx 

31,  x^-ax  =  bx  32.  ic2-f  ax= -6x 

33.  x(x-Q)  =  S-4:ix-\-2)  34.  7y-4(y-3)  =  3(4-y2) 

35.  Sn-n(n-S)  =  n(n-9)        36.  6(«-2)  =  4(w2-3) 


152 


BEGINNERS'  ALGEBRA 


111.  Product  of  two  binomials.  The  floor  space  of  the 
four  rooms  represented  in  the  diagram  (Fig.  37)  may  be 
computed  in  several  ways.  We  may  find  the  floor  space  of 
each  room  in  the  following  way: 

20  •30+20- 8+7- 30+7 -8 
=  600+160+210+56 
=  1026 


or  we  may  compute  the  whole 
floor  space  at  once: 

(20+7)  (30+8)  =27  •  38=  1026 


30 

8 

20 

7 

8 

Fig.  37 


Notice  how  the  multiplication  is  done: 


38 
27 


or  in  a  longer  form 


266 
76 


30+8 
20+7 
210+56 
600+160 


1026  600+370+56  =  1026 

Evidently  (20 +7) (30 +8)  =20  •  30+  20  •  8+7  •  30+7  •  8. 
So  also  for  the  floor  space  in  Fig.  38. 
The  area  room  by  room  is 

x^+bx+^+5  •  4  =  :x:2+9:\;+20 
The  area  of  the  whole  is 

The    multiplication  is    to  be 
carried  on  as  before: 

X  +  4: 

x+5 


5it+20 

x^+^x 
A;2+9ic+20 


^  5 

4 


Fig.  38 


PRODUCTS;  FACTORING;  EQUATIONS  153 

Since  we  write  from  left  to  right,  it  is  more  convenient  in 
algebra  to  begin  multiplying  at  the  left  and  put  down  the 
work  from  left  to  right: 

X  +  4:  %-Z 

x^-\-4x  x^—Sx 

f5^+20  +6^-18 

x'-{-3x+20  x^-^Sx-lS 

The  product  of  the  two  binomial  factors  is  made  up  of 
the  sum  of  the  products  of  every  term  of  one  factor  with 
every  term  of  the  other  factor. 

It  is  not  necessary  to  write  the  multiplication  down  in 
the  arithmetical  form  given  above.  The  work  can  be 
written  just  as  weU  thus: 

(x+5)(x-\-4:)=x{x-\-4d-\-5{x+4.) 
=  ^^2+4^+5:^+20 
=  x^-\-9x-\-20 
{x-d)ix+Q)=x(x-{-Q)-S{x-{-Q) 
=  x^-\-Qx-3x-lS 
=x^+Sx-lS 

EXERCISES 

Find  the  following  products:  ' 

1.  ix+7){x+9)  2.  (x+Q){x+2)  3.  {x-7)(x-Q) 

4.  {x-8){x-3)  5.  {x-\-9)(x-5)  6.  (:x;+ll)(:r-7) 

7.  (w+l)(w-6)  8.  (w-ll)(w-12)        9.  (/+15)(/+7) 

10.  (/+15)(/-7)         11.  (/-19)(/+3)  12.  (a+8)(a-5) 

13.  (a-8)(a+3)  14.  {x+a){x-{-b)  15.  (n-{-t){n+s) 

Can  you  discover  a  short  way  of  writing  the  product  down  at 
once  without  writing  the  intermediate  steps? 
{x+5){x+Q)  =  x--^nx-\-30 
Apply  where  possible  to  the  following: 

16.  {x+7){x+2)  17.  {x-5){x-S)  18.  (n-5)(w+7) 

.  19.  («-8)(«+4)         20.  (a-15)(a-3)  21.  (a-15)(a+4) 
11 


154 


BEGINNERS'  ALGEBRA 


22.  (a-l)(a+7) 
25.  (»+3)(jc-9) 
28.  (3;-6)(y+8) 
31.  {x-S)ix+4) 
34.  (r-8)(r-12) 
37.  {x-i){x+^) 
40.  (:r-3)(:r+13) 
43.  (a-6)(a+9) 
46.  (:i;+3)(:r+13) 
49.  81  •  83 

52.  ix^-x-S)(x-l) 

53.  {x^-2x-\-l){x+S) 

54.  {x^-Sx+2){x-S) 

55.  (a;2-x+4)(ic+2) 

56.  (a:2+:r-2)(ic-2) 

57.  We  have  seen  that 

{x+a)(x+b)  =  x'^-\-ax-\-bx-\-ab 
=  x^-{-{a+b)x+ab 


23.  (a+9)(a+3) 
26.  (a+10)(a+8) 
29.  (x-mx-^i) 
32.  ib-S)ib-\-10) 
35.  (r-8)(r+12) 
38.  (x-i)(x-i) 
41.  ix+i){x-i) 
44.  (a+6)(a-9) 
47.  (40+3)(40-2) 
50.  69  •  73 


24.  {x-S){x-Q) 
27.  (a-10)(a+8) 
30.  {x-\-7)(x-7) 
33.  (x^lXx-^i) 
36.  (r4-8)(r-12) 
39.  (j-6)Cv-ll) 
42.  (w-l)(w-3) 
45.  (x-ll)(:^+12) 
48.  (30+3)(30+5) 
51.  33-27 


X 

a 

X 

x^ 

ax 

X 

b 

bx 

ab 

b 

X  a 

Fig.  39 


58.     Show  that,  in  Fig.  40,  0=5 -^-5 -A 
Translate  into  algebraic  symbols,  using  data  on  figure. 


59.  Draw  a  figure  similar  to 
Fig.  39  to  illustrate  the  product 
{x  —  a){x  —  b). 

60.  Draw  a  similar  figure  to  S  = 
represent  {x  —  a){x-{-b). 

01.  Draw  a  similar  figure  to 
represent  {x—a){x-\-a). 


A 

3 
B  5 

C 

D 

Fig.  40 


PRODUCTS;  FACTORING;  EQUATIONS     155 

112.  Factoring  a  trinomial.  We  have  seen  that  the  prod- 
uct of  two  expressions  of  the  first  degree  is  an  expression 
of  the  second  degree.  We  may  reverse  the  point  of  view  and 
ask  the  question:  What  are  the  two  factors  whose  product 
is  a  given  quadratic  expression? 

What  are  the  factors  of  x^+12x-\-S5'^  We  have  noticed 
that  in  the  multiplication 

(r+3)(^+5)  =^24.(3^5)^.^3 .  5 

(1)  The  coefficient  of  the  x^  term  is  1. 

(2)  The  coefficient  of  the  x  term  is  the  sum  of  3  and  5. 

(3)  The  term  that  contains  no  x,  called  the  absolute 
term,  is  the  pi:oduct  of  3  and  5. 

These  facts  may  be  used  to  undo  the  multiplication  that 
gave  x^-\-Sx-\-15'.  15  is  the  product  of  the  absolute  terms 
of  the  two  factors,  and  8  is  the  sum  of  these  same  absolute 
^ terms.     Now  apply  this  to  x^-\-12x-\-3o. 

Thirty-five  is  to  be  regarded  as  the  product  of  two  unknown 
nimibers  whose  sum  is  12. 
Separate  35  into  two  factors  whose  sum  is  12: 

35  =  5 -7  and  5+7  =  12 
Hence  5  and  7  will  do,  and  x-\-5  and  x+7  are  the  desired 
factors;  that  is, 

x^+12x-{-36=-{x+5){x+7) 

In  most  cases  the  absolute  term  of  the  trinomial  has 
several  pairs  of  factors.  The  various  pairs  must  be  tried 
until  a  pair  is  found  that  works,  and  then  the  factors  will 
fall  apart  as  if  by  magic. 

.  x^-{-9x'\-20 

The  pairs  of  factors  of  20  are  2  •  10,  1  •  20,  and  4  •  5.  The 
2  •  10  pair  does  not  work.     The  4  •  5  pair  does  work. 

Hence  x^+9x+20=ix+5){x-\-4:) 


156  BEGINNERS'  ALGEBRA 

Consider  x^  —  5x-\-Q 

The  pairs  of  factors  of  6  are  2*3,   1*6;  neither  pair 
works,  but  the  pair  —  2  •  —  3  works. 
Hence  x^-5x-\-Q  =  {x-2)(x-S) 

Consider  x^—x  —  Q 

A  pair  of  factors  of  —6  whose  sum  is  —1  is  to  be  found. 

—  3  and  +2  is  such  a  pair 
Hence  x^-x-Q={x-Z){x-\-2) 

In  selecting  the  proper  pair  of  factors  of  the  absolute 
term  it  will  be  of  service  to  note  that : 

(1)  If  the  absolute  term  is  negative,  one  of  the  pair  must 
be  negative  and  one  positive.  How  can  you  determine 
which  is  to  be  negative? 

(2)  If  the  absolute  term  is  positive,  the  numbers  of  the 
pair  are  of  the  same  sign.  How  can  you  determine  which 
sign?  If  no  satisfactory  pair  can  be  found,  you  will  have 
to  say  that  you  cannot  factor  the  expression.  This  does 
not  mean  that  it  cannot  be  factored,  but  only  that,  as  far 
as  you  know  with  your  present  knowledge,  it  cannot  be 
factored. 

Practice  will  improve  your  skill  in  selecting  the  proper 
pairs  of  numbers. 

EXERCISES 

Factor  if  possible: 

1.  x24-7:«:+12  2.  n^+lSw+SO  3.  w2+5w+12 

4.  x^-bx+Q  5.  x^-7x-\-5  6.  a^+8x-\-15 

7.  l^-\-Sx-\-l  8.  a^+a+5  9.  x^-Qx+5 

10.  a'-^2a-S  11.  x^-^bx-Q  12.  w^-Tw-G 

13.  x^-Sx-28  14.  /2-I-6/-7  15.  7i^+7n-\-5 

16.  n2+7w+6  17.  P-Qt+7  IS.  .^2_7;c-18 

19.  p^+p -132  20.  /2-J-23/+102  21.  52-45+3 

22.  r2_ii;._60  23.  x"-5x-Si  24.  x^+lOx+24: 

25.  5^+75-60  26.  52-55+4  27.  /«+13/-300 


PRODUCTS;  FACTORING;  EQUATIONS 


157 


28.  ;c2+ 11^+24 
31.  22-9Z-36 
34.  x^+ix-ii 
37.  x'-2x-So 
40.  x-'-x-SO 
43.  n^-lOn-56 
46.  a^-a-12 
49.  ^2+10:^-56 
52.  x''-x-2 

Multiply: 
55.  {5-x){Q-\-x) 
58.  (9-{-x)i3-x) 
61.  (3-x)(5+a;) 
64.  (S-{-x){5-x) 
67.  (i-o;)  (i+:^) 
70.  (9+x){10-x) 
73.  (10-:x;)(12+:r) 

Factor: 
76.  7-6^-/2 
79.  60-llic-a;2 
82.  200-10x-:c2 
85.  90-23:c+x2 
88.  80-llrx;-:*;2 

91.  7o-2Sx-^x^ 
94.  100-29x+x2 


29.  w2-8w4-12 
32.  22-12Z+36 
35.  x^-{-Ux+24: 
38.  :\:2-lliC+30 
41.  a2+13a+12 
44.  a2+55o-56 
47.  x2- 15:^+56 
50.  x2+4:«;+3 
53.  x^-%x-\-i 

56.  (7-a;)(8-:»;) 

59.  (i-:r)(i+^) 

62.  {i-x){i-^x) 

65.  (2-x)(13+:c) 

68.  ii-xKi-x) 

71.  (4-:c)(f+x)^ 

74.  (5-rx:)(20+:r) 


77.  18-7ic-«2 
80.  60+17:«-Jc2 
83.  i-x-hx^ 
86.  80-2U+.r2 
89.  75-20x+:r2 
92.  100-25:t:+a;2 
95.  48-13:c-x2 


30.  >'2_  19^+48 
33.  x'--x-{-i 
36.  22  _  52  _  36 
39.  a;2-17.r+30 
.  42.  a2_|_9a+i8 
45.  62+11^,4-30 
48.  x'--x-dQ 
51.  ^2+26:c-56 
54.  x^+^,x-% 

57.  (7-:r)(5+a;) 

60.  (S-x)i5-x) 

63.  (8-:t:)(6+:^) 

66.  (3+:c)(ll+x) 

69.  (9-a;)(ll+:c) 

72.  (12-:*:)(4+a:) 

75.  (7-;c)(H-:^) 

78.  lS2+p-p^ 
81.  80-16:c-a:2 
84.  90+13:c-x2 
87.  i-ix+x^ 
90.  7o-]-l0x-x^ 
93.  100-15.r-:c2 
96.  2-x+x^ 


Supply  numbers  that  will  make  the  following  expressions  factor- 
able. If  more  than  one  such  number  is  possible,  supply  as  many 
as  you  can : 


97.  x^+?x-\-21 
100.  n^-4:n-\-? 
103.  ?-\-Ax-x^ 
106.  (  )-9x+x^ 


98.  0:2 -3:*:-? 
101.  x^+?x+7 
104.  w2+6w-(  ) 
107.  30+?x-«2 


99.  «2_|_?„_|_9 
102.  28+?^-:c2 
105.  x'-{-8x-\-{  ) 
108.  x^-?x-\-l5 


158  BEGINNERS'  ALGEBRA 

109.  15-?x-x^  110.  x'-i  );c+14  111.  (  )-\-5x-x^ 

112.  a:2+7a;4-(  )  113.  x^-{  ):c+18  114.  x^-(  ):c-18 

115.  x^-{  )x-SQ  116.  x^-{  )x+3G  117.  ?-\-Ux-x^ 

118.  (  )+x^-Sx  119.  a;2-(  ):x:+21  120.  x^-Sx+? 

Draw  the  graphs  of  the  following  expressions  of  the  second  degree. 
Let  the  unit  of  the  vertical  scale  be  one-half  the  unit  of  the  hori- 
zontal scale.     Note  the  points  where  the  graph  crosses  the  jc-axis. 

121.  x'-\-2x-S  122.  x^-{-2x-S  123.  a:=+2:r+l 

124.  jt;2-f2x+2  125.  x^-ix-^  126.  {x-2){x+l) 

127.  {x-3){x-2)         128.  8-2jc-;r2  129.  x'-{-x+l 

113.  Solution  of  equations  of  the   form   x~+mx-\-n  =  0. 

We  are  now  ready  to  solve  the  problem  proposed  at  the 
beginning  of  this  chapter  (Art.  94).  A  man  wishes  to  double 
the  area  of  an  8  by  12  rod  field  by  adding  to  one  end  and 
one  side  strips  of  equal  width.  How  wide  must  each  strip 
be?  The  equation  to  be  solved  is 
^2+205  =  96 

An  equation  of  this  type  may  be  solved  by  factoring  as 
follows : 

Put  in  form  5^ + 205  -  96  =  0 

Factor,  (5+24)(5-4)  =0 

Either  factor  put  equal  to  zero  will  give  a  root  of  the 
equation : 

s— 4  =  0  gives  5  =  4 
5+24  =  0  gives  5= -24 

Check  the  results  by  substituting  them  in  the  original 
equation. 

Both  4  and  —  24  are  roots  of  the  equation.  Can  both  be 
used  as  answers  to  the  problem  that  gave  rise  to  the  equa- 
tion?    Why? 

It  will  be  noticed  that  this  method  of  solving  equations 
of  this  form  applies  when  the  coefficients  m  and  n  are  such 
that  w  has  two  factors  whose  sum  is  ni.  Cases  in  which 
this  is  not  true  are  considered  in  Art.  178. 


PRODUCTS;  FACTORING;  EQUATIONS  159 

EXERCISES    AND    PROBLEMS 

Solve: 

1.  {x-4:){x-2)  =  0  2.  ix-2){x+S)  =  0  3.  (a+5)(a+8)=0 

4.  x^-{-5x+Q  =  0  5.  o2_|-a-6  =  0  6.  a;2+18=lLr 

7.  «2+24=llw  8.  :K2  =  a:+56  9.  0  =  ^2_f_4^_2i 

10.  w2  =  6«+18  11.  n24-36=15w  12.  x^-\-5x  =  Q 

13.  3;2+12>;  =  45  14.  y*  =  5^+14  15.  y^-y  =  72 

16.  The  length  of  a  rectangle  is  2  feet  more  than  its  width.  If 
its  area  is  63  square  feet,  find  its  dimensions. 

17.  If  5  feet  be  subtracted  from  two  opposite  sides  of  a  square, 
the  area  of  the  resulting  rectangle  will  be  25  square  feet.  Find  the 
side  of  the  square. 

18.  The  length  of  a  rectangle  is  3  feet  more  than  its  width.  The 
area  is  40  square  feet.     Find  length  and  width  of  the  field. 

19.  To  find  the  area  of  a  triangle  multiply  the  base  by  the  alti- 
tude and  divide  the  result  by  2.  Is  this  the  same  as  multiplying 
the  base  by  half  the  altitude?  As  multiplying  the  altitude  by  half 
the  base?  Why?  Express  the  rule  for  the  area  of  a  triangle  as  a 
formula. 

20.  Find   the  area  of  the  following 
triangles : 

(1)  Base  =12,  altitude  =  6 

(2)  Base  =15,  altitude  =  7 

(3)  Base  =16,  altitude  =  8 

(4)  Base  =  a,  altitude  =  5 

(5)  Base  =  a,  altitude  =  a— 3 

21.  The  base  of  a  triangle  is  4  feet  more  than  the  altitude.  If  the 
area  is  30  square  feet,  find  the  dimensions. 

22.  The  base  of  a  triangle  is  3  feet  less  than  the  height.  What  are 
the  dimensions  if  the  area  is  90  square  feet? 

23.  The  altitude  of  a  triangle  is  3  feet  less  than  the  length  of 
the  base.     If  the  area  is  27  square  feet,  find  the  dimensions. 

24.  The  altitude  of  a  triangle  is  10  feet  more  than  the  length  of 
the  base.     If  the  area  is  72  square  feet,  find  the  dimensions. 


160  BEGINNERS'  ALGEBRA 

25.  One  leg  of  a  right  triangle  is  2  feet  less  than  the  other.  Find 
the  dimensions  if  the  area  is  40  square  feet. 

26.  One  leg  of  a  right  triangle  is  7  feet  more  than  the  other. 
Find  the  other  leg  if  the  area  is  72  square  feet. 

27.  The  sum  of  the  base  and  altitude  of  a  rectangle  is  20  feet,  the 
area  is  91  square  feet.     Find  the  dimensions. 

28.  The  sum  of  the  base  and  altitude  of  a  triangle  is  21  inches. 
Find  both  if  the  area  is  45  square  inches.  Interpret  the  two 
answers. 

29.  The  difference  between  the  sides  of  a  rectangle  is  2  feet;  the 
area  is  99  square  feet.     Find  the  dimensions. 

30.  The  difference  between  the  base  and  the  altitude  of  a  tri- 
angle is  3  feet;  the  area  is  77  square  feet.  Find  the  base  and  the 
altitude. 

31.  The  sum  of  the  squares  of  two  consecutive  integers  is  841. 
Find  the  numbers. 

32.  Find  two  consecutive  integers  such  that  if  the  larger  is  sub- 
tracted from  the  square  of  the  smaller  the  result  is  19. 

33.  A  rectangle  is  3  times  as  long  as  it  is  wide.  If  2  feet  are  added 
to  the  length  and  2  feet  added  to  the  width,  the  area  is  increased  by 
36  square  feet.     Find  the  dimensions  of  the  rectangle. 

34.  A  rectangle  is  3  times  as  long  as  it  is  wide.  If  its  length  is 
decreased  by  4  feet  and  its  width  increased  by  2  feet,  the  area  will 
be  unchanged.     Find  its  dimensions. 

35.  (x-5)(x-Q)-{x-5){x-4:)  =  2 

36.  {x-3)ix-\-2)  -(x-4){x-\-Q)  =  27 

37.  (x-{-S)(x-4)-\-{x-S){x-2)  =  2 

38.  ix-\-5){x-l)-{-{x-2){x+Q)+25==0 

39.  (w-3)(«+8)-(»-6)(«+2)=24 

40.  3x2-(x-4)(a:-2)  =  12 

41.  4y2-(3y-l)(y+3)  =  68 

42.  ix-5){x-\-l)  =  7-{x-2){x-6) 

43.  50-(x4-2)(a:-3)  =  (JC-8)(x-3) 


PRODUCTS;  FACTORING;  EQUATIONS 


161 


114.  The   general   quadratic   trinomial.     We  have  been 
factoring  expressions  of  the  form 

in  which  the  coefficient  of  the  square  term  is  1.  A  ques- 
tion that  might  nattirally  arise  is:  Would  it  be  possible  to 
factor  an  expression  in  which  the  coefficient  of  the  square 
is  some  other  number  than  1 —  for  instance ,  5x^ + 7^  —  6  ?  To 
guide  us  to  the  best  way  of  untangling  the  factors,  if  there 
be  any,  we  must  first  consider  certain  multiplications. 


115.  Products  of  binomials. 

Multiply     3:x;+5  by  2iv+7 

Multiply 

3^+5  by  2x- 

Sx+5 

Sx-\-5 

2^+7 

2^-7 

6jt;2+10:^ 

6%2+10;t 

+21:\;+35 

-21^-35 

6x^+Slx+S5 

6^2-11^-35 

EXERCISES 

Find  the  following  products: 
1.  (3^+2)  (:^+l)  2.  (3x-l)(2;^-3) 

4.  (4x-3)(a;+2)  5.  {x-d){5x-8) 

7.  {3x+2){x-l)  8.  (3:c-l)(5:c+2) 


3.  (4:K+3)(:r-2) 
6.  (3jc-|-1)(2x+3) 
9.  (2a:-3)(7jc+2) 


Can  you  find  a  short  way  of  writing  the  product  at  once  without 
putting  down  the  intermediate  steps?    Apply  to  following: 

10.  {3x-5){5x-3)       11.  {2x-7)(9x-\-5)       12.  (4:*:- 5) (5:^-6) 
14.  (3-5x)(l+2x) 
17.  in-7){7n-l) 
20.  il-9y)i5-y) 
23.  (2w+7)(3w-2) 
26.  (s+7){s-S) 
29.  (y-3)(:y-7) 
32.  (n+l)(3+/j) 


13.  {2+5x){S-2x) 
16.  (6w-l)(3»-2) 
19.  {2y-6){3y+7) 
22.  {8-\-9x){5-2x) 
25.  (8/-3)(^+l) 
28.  (>'-10)(2>r-3) 
31.  (2n-S){Sn-2) 


15.  (3x4-7)(5;c+9) 
18.  (8a-3)(5a+4) 
21.  (x+2)(5x-7) 
24.  (3-«)(7+w) 
27.  (25 -3)  (7^+5) 
30.  (23'+7)(3'+9) 
33.  (2n+3)(2+») 


162  BEGINNERS'  ALGEBRA 

116.  Factors  of  quadratic  trinomial.  We  wish  to  factor, 
if  possible,  expressions  like  21a;^+29:j[:  — 10.  The  way  to 
untangle  the  factors  will  be  made  clear  if  we  look  over 
carefully  the  multiplications  of  the  last  article;  for  instance, 
Sx-\-5  by  2::^  — 7,  given  at  the  beginning  of  the  article. 

3^4-5 
2:j;-7 
It  is  seen  that  6x^  in  the  result  is  the  product  of  3x  and  2x, 

—  35  in  the  result  is  the  product  of  5  and  —7, 

—  lljc  is  the  algebraic  sum  of  the  cross  products,  2x  and  5, 
and  Sx  and  —7,  that  is,  of  10;c  and  —  21r!t;. 

Sx  +5 
2x  -7 

-21a: -35 


-llx 

Use  these  facts  in  factoring  21:^2+29^-10. 
Try  pairs  of  factors  of  21  and   —10.     For  instance,  try 
3  and  7  for  21,  —5  and  2  for  —10.     Find  whether  the  sum 
of  the  cross  products  is  29. 

3^-5 
7^+2 

-35jc 
+  6jc 


-29jc 

This  does  not  work,  as  the  coefficient  should  be   +29 
instead  of  —29. 

Again  try  Sx-\-5 

7x-2 
+35 
-  6 


+29 

This  works.     Hence  the  factors  are  3^+5  and  1x^2. 
Therefore 

21rc2+29ic- 10=  (3ic+5)(7x-  2) 


PRODUCTS;  FACTORING;  EQUATIONS 


163 


Factor: 
1.  10a;2+19:r+6 
4.  10x2+4a;-6 
7.  2x^-{-nx+^ 
10.  6+5fl-6a2 
13.  2/^2+// -28 
16.  2 -oh -31  f- 
19.  3+ 7a; -20x2 
22.  3 -19a: -14x2 
25.  5+14«-3;i2 
28.  243^2  _29>'- 4 
31.  12x2-23^+10 
34.  3.T2+10X+3 


EXERCISE    I 

2.  10x2 -X -21 
5.  15x2+16x+4 

8.  2x2+llx-8 
11.  3x2+7x-6 
14.  10-19^-15/2 
17.  4/2+/ -14 
20.  18x2 -9x -5 
23.  7x2 -X -8 
26.  9>;2-44>'-5 
29.  63'2+7>'-3 
32.  7+10«+3«2 
35.  9/2+16-24/ 


3.  10x2+16x+6 
6.  15x2 -4x -4 
9.  4c2+23c+15 
12.  4-5x-6x2 
15.  2/2+3/-2 
18.  5x2+23x-10 
21.  5-33x+18x2 
24.  8«2+;j-9 
27.  5-8a-4a2 
30.  3+196-14&2 
33.  x2+43x+390 
36.  7«2_15m-18 


EXERCISE    II 


Insert  numbers  where  indicated  so  that  the  following  expressions 
can  be  factored: 


1.  2x2 -5x+? 

4.  5x2+?x+12 

7.  7+?x+3x2 

10.  ?-llx+7x2 

13.  18+?x-5x2 


2.  3+?x-2x2 
5.  ?-4x+3x2 
8.  7x2 -4x-? 

11.  9x2-25x+? 

14.  ?-x-15x2 


3.  5x2 -4x-? 
6.  10+?x-3x2 
9.  2x2-?x+35 

12.  5x2-19x+? 

15.  15x2 -?x+2 


EQUATIONS 

Solve  the  following  equations  and  problems  by  the  method  of 


factoring: 

1.  2x2+x-21  =  0 
3.  2x2+3x  =  35 
5.  5x2+14  =  37x 
7.  2x2-6x  =  7x-15 


2.  3x2 -16x -35  =  0 
4.  7x2=15-32x 
6.  3x2  =  25x-28 
8.  3x2+8x  =  3x+28 


9.  9x2=(3x+7)(x+2)-19    10.  6x2-(2x+3)(x+3)=4 

l-2x    x(x-2)     X 
2  3       ~2 


x(x-4)  .  l_x-l 
^^'        6         -6       4 


12. 


104  BEGINNERS'  ALGEBRA 

13.  (2:c-5)(3:»-5)-(:k-4)(3x-1)  =  9 

14.  (3:x:-1)(3jc+4)-(3C+4)(2^-5)  =  17 

15.  (3x-l){x-5)-(2x-\-5){x-l)=30 

16.  {5x-4:)ix'-l)  -(Sod+2){x-Jr2)  =  SO 

17.  The  length  of  a  rectangle  is  5  feet  less  than  3  times  its  width. 
If  its  area  is  78  square  feet,  find  its  dimensions. 

18.  The  length  of  a  rectangle  is  1  foot  more  than  4  times  the 
width.     Find  the  dimensions  if  the  area  is  39  square  feet. 

19.  The  base  of  a  triangle  is  1  foot  more  than  twice  the  altitude. 
If  the  area  is  52i  square  feet,  find  the  dimensions. 

20.  The  altitude  of  a  triangle  is  1  foot  less  than  3  times  the  base. 
Find  the  dimensions  if  the  area  is  22  square  feet. 

21.  The  area  of  a  certain  rectangle  is  59  square  feet  more  than 
the  area  of  a  certain  square.  If  the  length  of  the  rectangle  is  7  feet 
less  than  twice  the  side  of  the  square  and  its  width  is  1  foot  more 
than  the  side  of  the  square,  find  the  dimensions  of  each. 

22.  The  sum  of  the  areas  of  a  square  and  a  triangle  is  58  square 
feet.  The  base  of  the  triangle  is  1  foot  longer  than  the  side  of  the 
square  and  its  altitude  1  foot  more  than  twice  a  side  of  the  square. 
Find  the  dimensions  of  each. 

23.  The  units  digit  of  a  certain  number  is  2  more  than  the  tens 
digit.  The  number  itself  is  1  less  than  4  times  the  square  of  the 
tens  digit.    Find  the  number. 

24.  Find  three  consecutive  integers  such  that  the  square  of  the 
first  added  to  the  product  of  the  other  two  is  67. 

25.  If  the  area  of  a  certain  rectangle  is  subtracted  from  3  times 
the  area  of  a  certain  square,  the  result  is  28  square  feet.  The  base 
of  the  rectangle  is  7  feet  more,  and  its  altitude  is  6  feet  more  than 
a  side  of  the  square.     Find  the  dimensions. 

26.  Divide  17  into  two  parts  so  that  the  product  of  the  two  parts 
is  21  more  than  the  square  of  the  smaller  part. 

27.  The  tens  digit  of  a  certain  number  is  2  more  than  the  units 
digit.  The  number  itself  is  5  more  than  twice  the  product  of  the 
digits.    Find  the  number. 


PRODUCTS;  FACTORING;  EQUATIONS  165 

117.  Special  forms.     Square  of  a  binomial.     If  the  two 

binomial  factors  are  equal,  the  product  takes  on  a  special 
form  of  considerable  importance. 
For  instance,  (^+3)  (^+3) 

^+3 
:r+3 

+3:x:+9 


The  special  characteristics  of  this  form  stand  out  more 
clearly  in  a  more  general  case: 

a^-\-ab 


a2+2a6+62 
that  is,  {a-\-b){a+b)  =  (a+by  =  a^-h2ab+b' 

Let  the  pupil  verify  by  using  various  numbers  for  a  and  b. 

Before  reading  any  further,  discover,  if  you  can,  the  special 
features  of  the  identity  given  above.  Do  the  same  character- 
istics appear  in  {a  —  by=  ?  What  difference  do  you  see  in 
the  two  illustrations  (a-\-by=?  and  (a  —  by=?  It  is  well 
to  express  the  identity  in  words. 

The  square  of  the  sum  of  two  numbers  equals  the  sum 
of  the  squares  of  the  ntunbers  plus  twice  their  product. 

Make  a  similar  statement  for  the  case  (a  — by. 

This  statement  is  to  be  regarded  as  a  formula  or  blank 
form  which  is  to  be  filled  in: 

(n+of=n^2no-o' 

Fig.  42 
{2x+3y  =  i2xy  +  2 -2% -3  +  3^  =  ix^-Jt-12x-^9 


166 


BEGINNERS'  ALGEBRA 


EXERCISES 

Write  out  the  products  by  use  of  the  formula: 


1.  (x+5)2 

2.  (y+l)2 

3.    (^+11)2 

4.  (w-6)2 

5.  (:v-9)2 

6.  (4-w)2 

7.  (2a+3)2 

8.  (3w-2)2 

9.  (5;c+l)2 

10.  (3-7/)2 

11.  (r-ty 

12.  (2r-30' 

13.  i5r-2xy 

14.  (2a+96)2 

15.  (20+1)2 

16.  (30-1)2 

17.  4P=  (40+1)2 

18.  332 

19.  612 

20.  522 

21.  482 

22.  99« 

23.  882 

24.  692 

25.  182 

26.  (1+3)2 

27.  (1-4) 

-(■4> 

29.(5-^-)2 

30.  (:.  +  |y) 

31.  (4-!.). 

32.  (a+l,y 

-  (?-!')• 

34.  Plot  (.r -2)2 

■  for  values  of  x  from  x  =  • 

-5toa;  =  +5. 

35.  Plot  (2-xy 

'  for  values  of  x  from  x  =  ■ 

-5to:«=+5. 

36.  Plot  (:^:;+ 1)2  for  same  values. 

37.  Plot  ( -3)2  for  same  values. 

38.  Plot  a:2-2jc+1  for  same  values. 

39.  Plot  .x2+4jc+4  for  same  values. 

40.  What  is  the  relation  of  these  graphs  to  the  ic-axis? 

41.  Plot  (x  — 2)2  for  same  values. 

42.  What  is  the  difference  between  -{x-2)^  ix-2y-,  i2-xy? 
What  is  the  difference  between  their  graphs  ? 

118.  Special  forms.    Factoring   of   a   trinomial   square. 

Factoring  4a;2+12a;+9  by  the  crisscross  method,  we  get  the 
factors  (2x+3)(2:r+3).  The  factoring  might  be  done  more 
quickly  if  one  noticed  that  the  expression  was  a  trinomial 
square,  that  is,  the  form  obtained  as  the  product  of  two 
equal  binomials.     It  is  rather  easy  to  recognize  such  a  form. 

a2+2a6+62  =  (a+6)2 


PRODUCTS;  FACTORING;  EQUATIONS  167 

(1)  The  expression  on  the  left  has  three  terrtis. 

(2)  Two  terms  are  positive  and  squares. 

(3)  The  third  term  is  twice  the  product  of  the  square 
roots  of  the  square  terms. 

a^  is  the  square  of  a.     We  also  say  that  a  is  the  square  root 

of  a\ 

9  is  the  square  of  3 

3  is  the  square  root  of  9 

One  of  the  two  equal  factors  of  a  number  is  the  square 

root  of  that  number. 

So  in  4;t;2+12x+9 

the  two  end  terms  are  squares,  their  square  roots  being 

2x  and  3 
The  third  term  is  twice  the  product  of  these  two  roots, 
2-2;c-3  =  12x 
therefore  4:^2+12^^+9  =  (2^+3)^ 

The  trinomial  square  is  of  so  much  importance  in  mathe- 
matics that  it  is  well  to  become  very  familiar  with  it. 

EXERCISES 

Factor  when  possible: 

1.  x^-{-Qx-\-9  2.  x^.-{-2x-\-l 

3.  x2-2:c+l  4.  x\+Sx-\-m 

5.  ^2- 16^+64  6.  x?+ 12^-36 

7.  n2+12»+36  8.  n^-4n+4i 

9.  9-6w+w2  10.  ^-3/+9 

11.  A^-UA-\-4Q      ^  12.  x2+x+i 

4 

13.  f2+4/+6  14.  x'-\-^x-{-~ 

15.  4x«-13x+9  16.  4/+20y+2r3 

17.  /24- 121 -22/  18.  a;2-|a;+i 

19.  x^-^x+^  20.  9n2+25-30n 

5       2o 


168 


BEGINNERS'  ALGEBRA 


21.  a2-13a+36 

22.  x^-\-y^+2xy 

23.  4j2+1+45 

24.  4:X^-\-l2xy-\-9y^ 

25.  25n'+lQ-20n 

26.  81a2+25&^+90a6 

27.  9x'-2x+l 

^^•25+5'^^ 

29.  :x;2-l^+i- 

30.  :x:2-^:«-? 

12       12 

3       9 

Supply  the  terms  that  will  make  the  following  expressions  tri- 

nomial squares: 

31.  w24-2w+(    ) 

32.  w2+18m+(    ) 

33.  c2+(    )+9 

34.  a^+2ab+i     ) 

35.  l+(    )+4x2 

36.  .t2-16a;H-(    ) 

37.  x^'-2x-{-i    ) 

38.  w2-f(    )+49 

39.  a;2_i0;x;+(    ) 

40.  4«2_l2;i+(    ) 

41.  9x^+Qxy-{-{    ) 

42.  64c2-lG6c+(     ) 

43.  25a2+(    )-30a 

44.  4c5+4c(i-(    ) 

45.  16w2+25w2-(    ) 

46.  9jt;2+12:^+(    ) 

47.  49^:2+ (     )+25/ 

48.  x^+Sx+{    ) 

49.  x'-5x-\-i     ) 

50.  4ic2+6:c+(    ) 

51.  Plotic2-4:*;+4 

52.  Plot9-6:«+ic2 

53.  Plot  x''+Qx+9 

54.  Plot  1+2^+^2 

119.  The  product  of  the  sum  and  difference  of  two  num- 

bers.    In  the  multiplication 

o 

x^-Sx 

+Sx-9 

x^         -9 

the  term  of  first  degree  drops  out,  as  the  sum  of  the  cross 
products  is  zero.  The  general  formula  for  this  type  of 
multiplication  is 

{a+b)(a-b)=a^-b^ 
Let   the   pupil  verify  by  putting   numbers  in  place  of 
a  and  b. 


PRODUCTS;  FACTORING;  EQUATIONS 


169 


The  verbal  statement  is:  The  product  of  the  sum  and 
difference  of  two  numbers  equals  the  square  of  the  first 
number  minus  the  square  of  the  second. 

It  is  a  very  important  identity,  and  is  used  frequently 
in  mathematics.  Keep  in  mind  that  it  is  true  for  any 
values  that  may  be  assigned  to  the  letters. 


(7+5)(7-5) 

72-52 

12-2 

49-25 

24 

24 

EXERCISES 


Apply  the  identity  given  above  to  the  following  exercises: 


1.  (x-2)(x-{-2) 
3.  (20-1)  (20+1) 
5.  57  •  63 
7.  {x-7){x+7) 
9.  {5-x)i5-\-x) 
11.  (i-^xXi+x) 
13.  (/-16)(/+16) 
15.  23  •  17 
17.  (5 -10)  (5+ 10) 
19.  (9-w)(9+w) 
21.  {S-\-2n)(S-2n) 
23.  (2a+36)(2a-36) 
25.  {x-mx+%) 
27.  {Sx+7a){Sx-7a) 
29.  {x-\-3a)(x-Sa) 


2.  (x-l){x-^l) 

4.  (40 -3) (40+3) 

6.  48  •  52 

8.  {x-Q){x+6) 
10.  {ix-y){ix+y) 
12.  (ll-*)(ll+a;^ 
14.  98  •  102 
16.  (a-9)(a+9) 
18.  (6-y)(6+y) 
20.  (x-iXx+i) 
22.  {Sx-\-d){Sx-5) 
24.  199  •  201 
26.  (2^-5)(2a;+5) 
28.  {x-ia)(x-^ia) 


30.  (w-5/)(«+5/) 

31.  Evaluate  jc^— 4  for  all  integral  values  of  x  from  it;=  —  4  to 
x=  +4  and  plot  on  cross-section  paper. 

32.  PlotO-:^^ 

33.  Evaluate  x^+4:  for  all  values  of  x  from -3  to  +3,  and  plot 
on  cross-section  paper. 

34.  Note  the  points  where  these  graphs  cross  the  :r-axis. 
12 


170  BEGINNERS'  ALGEBRA 

120.  Factoring    the    difference    between    two    squares. 

Reading  the  identity  of  the  last  article  backward, 

a2-62=(a+6)(a-6) 
we  have  an  easily  recognized  form  that  can  always  be 
factored;  namely,    the    difference    between    two    squares. 
Why  is  it  so  called  ? 

Qn^— 49  is  of  this  type 
hence,  9n^  -  49  =  (3w+7)  (3w  -  7) 

The  identity  stated  in  words  is: 

The  difference  between  the  squares  of  two  numbers 
equals  the  product  of  the  sum  of  the  two  numbers  and  the 
difference  between  the  two  numbers. 

EXERCISES 


Factor: 

1.  x'-lQ 

2.  j2_36 

3.  »2-9 

4.  y2_25 

5.  49 -x^ 

6.  16 -«2 

7.  y^-1 

8.  x^-\ 
4 

9.  1-x^ 

10.  l-9w2 

11.  64»2_25 

12    x«-i 
•         25 

13.  4x2-81 

14.  a^-l 

15.  ^^2+36 

16.  n'-Sl 

17.  121ic2-144 

'''  ^'-u 

19.  n^-P 

20.  x^-4y^ 

21.  52-42 

22.  412-392 

^-  (If-& 

24.  ^-y^ 

16    25 

25.  522-482 

26.  632^472 

27.  272-232 

EQUATIONS   AND   PROBLEMS 

Solve  the  following  equations: 

1.  (n-3)(w+3)=0  2.  (2w-7)(2»+7)=0 

3.  n;'-25  =  0  4.  4x^-49  =  0 

5.  w«=144  6.  9a2  =  49 

7.  25*2-169=0  8.  196=  16a;* 


PRODUCTS;  FACTORING;  EQUATIONS  171 

9.  What  must  be  the  edge  of  a  cube  so  that  the  surface  of  the 
cube  shall  be  24  square  inches?     Is  there  more  than  one  answer? 

10.  The  width  of  a  certain  rectangle  is  f  of  the  length.  If  the 
area  is  120  square  feet,  find  the  dimensions. 

11.  The  altitude  of  a  certain  triangle  is  f  of  its  base.  Find  the 
dimensions  if  the  area  is  150  square  inches. 

12.  The  length  of  a  certain  rectangle  is  6  feet  more  than  its  width. 
If  its  length  is  diminished  by  4  feet  and  its  width  is  multiplied  by  3, 
its  area  will  be  increased  by  8  square  feet.     Find  its  dimensions. 

13.  The  square  of  a  number  increased  by  15  equals  184.  What 
is  the  number?     Is  there  more  than  one  answer  to  this  question? 

14.  Three-fourths  of  the  square  of  a  certain  number  is  108.  Find 
the  nimiber. 

15.  If  25  is  added  to  ^  of  the  square  of  a  certain  number,  the 
result  is  150.     Find  the  number. 

16.  If  1  is  subtracted  from  the  square  of  a  certain  nimiber  and 
the  remainder  divided  by  4,  the  answer  will  be  the  same  as  if  21 
had  been  subtracted  from  the  square  of  the  number  and  the  result 
divided  by  3.     Find  the  number. 

Many  equations  that  do  not  appear  at  first  sight  to  be  of  the  kind 
considered  here  may  be  reduced  to  this  standard  form: 

17.  5:;c2 -20  =  ^2 -16 

18.  a;2-4(2-A;)=24-x(a;-4) 

19.  x{x-{-^)  -75  =  Qx-2xix+l) 

20.  i'ix-3){x-2)-ix-S)ix-S)  =  o7 

21.  2x-x{x-4:)  =  lS-x{Sx-Q) 

■    22.  {3x-2)(2x-5)  -(x-20){x-\-l)  =  75 
23.  ^x^-x{x+15)  =  S2-\-x(5x-l5) 


2j2 


24. 

i-i=i+8 

25. 

X'          2x2+3 
3+^"     5 

26. 

352-41  =  39- 

27. 

98-8w2  =  0 

28. 

75/2  =  27 

29. 

x'  =  25a' 

30. 

7i»;2-252o2  =  o 

172  BEGINNERS'  ALGEBRA 

121.  Summary  of  important  identities.  In  the  preceding 
articles  we  have  considered  several  important  identities: 

(1)  a{b-]-c)=ab-]-ac 

(2)  {a+b)(a-b)=a^-b^ 
(x-}-3){x-3)=x^-9 

(3)  {a+by  =  a^+2ab-\-b^ 
ix-\-5y=x^+10x+25 
{a-by  =  a?-2ab+b^ 
(r^-5)2  =  ::k;2-10^+25 

(4)  {x+a){x+b)=x^^{a-]-b)x^-ab 
{x+2){x-h)=x''-?>x-\0 

(5)  {mx + a)  {nx  -\-b)=  mnx^  +  (aw  -h  bm)  x-\-ab 

{2x-S)iSx+5)=Qx^+x-15 

In  each  identity  there  are  two  forms  of  the  same  number, 
a  product  or  factored  form  and  a  sum  or  expanded  form. 
Tell  which  is  the  product  form  and  which  the  sum  form  in 
each  case.  Both  forms  have  important  uses.  One  form 
is  to  be  used  for  one  purpose.  Another  situation  may 
require  the  other  form.  It  is,  therefore,  important  for  you 
to  be  able  to  change  an  expression  from  one  form  into  the 
other  quickly  and  correctly;  that  is,  to  expand  an  expression 
or  factor  an  expression.  A  product  form  can  always  be 
expanded.  The  reverse  process,  factoring  a  sum  form,  can 
be  done  in  certain  cases  only. 

122.  Uses  of  the  identities.  These  identities  may  be 
used  for  three  different  purposes:  (1)  to  change  compli- 
cated expressions  into  less  complicated  forms;  (2)  to  change 
an  expression  into  a  form  that  can  be  used  for  a  given  pur- 
pose ;  (3)  to  reduce  the  amount  of  work  in  making  numerical 
calculations. 

(1)  To  illustrate  the  first  use^  take  the  expression, 
{x-2Y+{x+2){x-Z) 


PRODUCTS;  FACTORING;  EQUATIONS  173 

A  knowledge  of  the  identities  enables  us  to  write  out  the 
expansion,  or  multiplications  indicated,  at  sight, 

which,  upon  collecting  terms,  becomes 

A  similar  use  of  these  identities  will  come  to  light  in  our 
study  of  fractions  in  a  later  chapter. 

(2)  To  illustrate  the  second  use,  notice  that  the  factored 
form  is  used  in  solving  equations. 

In  solving  2x''-bx+2  =  0 

we  factor  ^  2^2-5^ -f2  =  0 

getting  {2x-l){x-2)=0 

(3)  To  illustrate  the  third  use,  notice  that  the  labor  of 
nimierical  calculation  can  often  be  greatly  reduced  by 
factoring. 

For  instance,      46  •  37+46  •  15  =  46(37+15) 

=  46-52 
Compute  both  ways  and  compare  the  number  of  figures 
used. 

So  also  in  a  case  like 

482-352=  (48+35)(48-35) 
=  83-13 
Compare   both   ways   of  making   the   computation   and 
determine  whether  the  factoring  method  does  really  lessen 
the  amount  of  work. 

It  is  possible  to  use  expanding  in  the  same  way: 
932=  (90+3)2 
=  8100+540+9 
=  8649 
19 -21  =  (20-1)  (20+1) 
=  400-1 
=  399 
Such  use  is  rather  limited  and  is  hardly  worth  while,  for 
in  most  cases  the  ordinary  way  of  computation  is  simpler 
•and  shorter. 


174  BEGINNERS'  ALGEBRA 

123.  Products  of  several  factors.  When  three  or  more 
factors  are  to  be  multiplied  together,  they  may  be  multi- 
plied in  any  order: 

3- 5 -7  =  15 -7  =  105 
or  21-5  =  105 

or  3-35  =  105 

EXERCISES 

Find  the  following  products  in  more  than  one  way.  Have  you 
any  choice  in  the  order  used? 

1.  9  -  6  .  5  2.  25  •  14  •  8 

3.  x{x-4){x-n)  4.  Sxix-5){x-^5) 

5.  xKx-l){x+2)  6.  SxKx-\-7)(x-3) 

7.  7x{S-x){S-{-x)  8.  4:cH:v+7)(:x;-2) 

9.  9:^(5 -:c)(:c+6)  10.  (x+10){x-S)x 

11.  {x-n)x{x+5)  12.  {x-\-2)xKx-2) 

13.  S{x+7)x'{x+7)  14.  5x(2:»:-l)  (2:^+7) 

15.  3:^(3x-l)(3:x4-l)  16.  2:«(:x:-l)  (3:^+4) 

17.  2a;2(2:«-l)(3ic+2)  18.  3:^(2:^ -3)  (2:x; -3) 

19.  5x2(1 -r«)  (3+2:*;)  20.  4x2(2x-l)(x-3) 

21.  Qx{S+x)a-7x)  22.  3x(l -x)(7+2x) 

124.  Possibility  of  factoring.  As  has  been  said,  an 
tmexpanded  expression  can  always  be  expanded.  But  it 
is  not  always  possible  to  factor  a  given  expression.  And 
it  is  just  as  important  to  know  that  a  given  expression  can- 
not be  factored  as  it  is  to  know  that  it  can  be  factored. 
You  must,  then,  become  so  thoroughly  acquainted  with 
factorable  types  that  you  can  recognize  them  almost  at  a 
glance.  The ,  factorable  types  we  have  discovered  so  far 
in  this  book  are: 

(1)  the  common  factor  form         ab-\-ac 

(2)  the  difference  of  two  squares  a^—b^ 

(3)  the  trinomial  square  a^+2ab-\-b^ 
(1)  the  trinomial  quadratic  ax^-{-bx-{-c 


PRODUCTS;   FACTORING;  EQUATIONS  175 

Types  1,  2,  and  3  are  always  factorable.  Type  4  is  fac- 
torable only  in  certain  special  cases,  depending  upon  the 
values  of  a,  6,  and  c. 

There  are  many  other  types  of  expressions  that  can  be 
factored,  but  these  are  sufficient  for  our  present  needs. 

To  factor  a  given  expression : 

1st.  Find  out  whether  the  expression  belongs  to  one  of  the 
types  with  which  you  are  acquainted. 

2d.  If  it  does,  it  is  readily  factored. 

3d.  If  it  does  not,  it  is  not  factorable  by  the  methods 
with  which  you  are  acquainted. 

The  work  may  be  checked  by  multiplying  the  factors 
together.     This  will  answer  in  most  cases. 

Another  method  of  checking  is  the  substitution  of  some 
convenient  number  for  the  letter: 

Check  by  substituting  10  for  x: 


102+3-  10-70 

100+30-70 

60 


(10+10)(10-7) 

20-3 

60 


125.  Several  factors.     Many  expressions  have  more  than 
two  factors. 
Illustration: 

Factor  x^-x^-20x 

^  is  a  factor,  as  it  is  in  every  term. 

jcS-x^-20x  =  xix^-x-20) 

Then  factor  r\;2_^_20 

x^-x-20  =  {x-5){x-\-4:) 

Therefore         x^-x^-20x  =  x{x^-x-20) 

=  x{x-5){x+4:) 

The  expression  has  been  factored  into  its  prime  factors. 


176 


BEGINNERS'  ALGEBRA 


Factor: 
1.  x^+x^-12x 
4.  24a-2a2_a3 
7.  83^'+ 14^2-153; 
10.  Sx^+27x^-Q6x 
13.  lOx^-Qx^-Ux 
16.  24aj4-3x2-21:^3 


EXERCISE   I 

2.  2x3_6^2_56^  3,  a<-12a3+32a2 

5.  2x^-\-18x^-72x        6.  8x4-2x'-15ic2 

8.  16a3+4a2-6a  9.  3/2-2/3-8/* 

11.  24^:3- 42:^2  _j_  9^  12.  33'2+103;3-8y* 

14.  5/-33>'3-14>'2  15.  7jc4-21ic3+28x2 

17.  8:x;4-17a;3+9:r2  18.  15w-48w2+9w3 


Expand  and  check: 
1.  (:^-3)(^+4) 
4.  (x-3)(x-3) 
7.  (x-5){x-{-Q) 

10.  ix-5){x-5) 

13.  (2^-l)(2x-3) 

16.  (5-x)2 

19.  (ic-4)(2x-l) 

22.  (3;c4-4)2 

25.  (3x-2)(2x-3) 

28.  (3-:r)(5-2x) 

31.  (l-5w)(l+5w) 

34.  56  •  64 


EXERCISE   II 

2.  {x-S)ix-4) 

5.  {x-\-S){x+3) 

8.  (r«-5)C^-9) 

11.  (:r-8)(:r+4) 

14.  {5-x){7-\-x) 

17.  (5-.t)(5+:x:) 

20.  (2x-l)2 

23.  (3x+4)(3^-5) 

26.  (6-2:c)(7+2:r) 

29.  (l-3x)(l-5x) 

32.  23  •  17 

35.  732 


3.  {x-]-S){x-4:) 

6.  (:*;+4)2 

9.  (;x;+5)2 

12.  (2iC-l)  (2:^+1) 

15.  (x-7y 

18.  {2x-5){dx-4:) 

21.  (3.t;-l)(3a;+l) 

24.  {Sx-5){Sx+5) 

27.  (6-2:r)2 

30.  (l-3a)2 

33.  38  •  42 

36.  932 


Factor  and  check: 
1.  :k2_25 
4.  p^-2p-{-l 
7.  62+16 
10.  n^-^n+l 
13.  64-9w2 
16.  2ab^-ia'-b 
19.  5:c2-2:i;-3 
22.  2jt;2+9jc-5G 
25.  4^2+6^  _7o 


EXERCISE    III 

2.  n'^-n 

5.  4x^-8x^ 

8.  p^-\-2p-d2 

11.  Sax''-\-7bx^ 

14.  3x4-3x3-36x2 

17.  6x2-13x+6 


3.  a2+2a-80 
6.  9m2_36/2 
9.  X2-143C+49 
12.  /2-/-90 
15.  2ax— 6a>' 
18.  6x2-5:«+6 


20.  12ax+20ax2-32ax3  21.  10x2-160 
23.  2x2-23x4-56  24.  3x2-27 

26.  5x2 -20x  27.  5.r»-20x 


PRODUCTS;  FACTORING;  EQUATIONS  177 

28.  2x2-17x+35  29.  81-121:x;2  30.  3:«2_7^_20 

31.  5Qx^-9x^-2x  32.  1-144:»;2  33.  7:^2-21^ 

34.  7x^-Q3x  35.  2x3-24x2+90:^  36.  2x^+3x2 -27x 

37.  6x3-14x2+8x  38.  9x'-39x2-30x  39.  12x3 -21x2 -45x 

40.  6x3+28x2 -lOx 

Use  factoring  in  the  following  exercises: 

41.  Find  the  value  of  75^  -25^. 

42.  Find  the  value  of  323^  -277^. 

43.  Find  the  value  of  35(43^  -37^). 

44.  Find  the  value  of  17  •  81^  - 17  •  69^. 

45.  Factor  Ta^-wr^.  Evaluate  the  result  when  a  =10,  r  =  4; 
when  a  =12,  r=3. 

46.  Draw  two  circles  with  the  same  center,  and 
with  radii  2^  inches  and  2  inches.  What  is  the  area 
between  them?     (Fig.  43.) 

47.  Draw  two  circles  with  the  radii  3  inches  and  2 
inches  but  so  that  the  circles  touch  each  other  on  the 
inside  of  the  larger  one.  Find  the  area  of  the  crescent- 
shaped  figure.     (Fig.  43.) 

48.  What  will  be  the  cost  of  laying  a  4-foot  concrete  walk  around 
a  circular  tower  20  feet  in  diameter,  at  90  cents  a  square  yard? 

The  area  of  the  curved  surface  of  a  cylinder  (Fig.  44)  is  the 
product  of  the  altitude  and  the  circumference  of  the  base.  If  r  is 
the  radius  of  the  base  and  h  is  the  height  of  the 
cylinder,  the  lateral  area  =  27rr/f.  The  area  of  each 
end  is  7rr2.  The  total  area,  then,  is 
27rr^+27rr2 

49.  What  is  the  total  area  of  a  cylinder  if  A  =  3  and 
r=2?/?  =  7andr  =  5? 

50.  What  is  the  area  of  the  total  surface  of  a 
cylindrical  box   14  inches  in  diameter   and   8   inches   high? 

51.  Allowing  nothing  for  overlapping,  how  much  tin  is  used  in 
making  a  can,  the  diameter  of  the  base  being  6 . 2  inches  and  the 
height  being  7 . 5  inches? 


178  BEGINNERS'  ALGEBRA 

52.  What  will  it  cost  to  cement  the  walls  and  floor  of  a  cylindrical 
silo  30  feet  high  and  15  feet  in  diameter  at  75  cents  a  square  yard? 

53.  Plot  x^-^.  54.  Plot  x^-9. 
55.  Plot  4:^2-25,                      56.  Plot  4-:x;2. 

126.  Miscellaneous  equations  and  problems.  Solve  and 
check : 

1.  a:2  =  121  2.  x^-7x-\-Q  3.  5:;c2  =  70jc 

4.  Sx^  =  5x  5.  ax^=bx  6.  a:2+2jc=63 

7.  /2_/_30  =  0  8.  3w24-7w  =  6         9.  n''-^n=^5 

10.  m^-10m  =  0  11.  a;2+32  =  12:«      12.  (jc-3)(:r+4)=0 

13.  (jc-2)(x4-3)x  =  0  14.  xix-7)(x-l)  =  0 

Note.  Observe  that  when  there  are  three  factors  containing  the 
unknown  there  are  three  roots  to  the  equation.     Why? 

15.  3(jc+5)(^-3)  =  0     16.  x'+dx^-Qx  =  0     17.  2n'-l0n^  =  28n 

18.  Plot  the  expression  x^  —Sx^-{-2x.  Note  the  points  where  the 
curve  cuts  the  rjc-axis.     Solve  the  equation  x^  —Sx'^-\-2x  =  0. 

19.  Solve  the  equation  ^^2+ or— 20  =  0.  Plot  the  expression 
x^-{-x-20. 

20.  Plotx3-9:\:;  solve  :k3-9x  =  0. 

21.  Solve  it:3-4a;2  =  0;    plot  x3-4ic2. 
Solve: 

22.  (/+ 10)2  =16/2 

23.  144=^w[48-4(w-l)] 

24.  ll(lH-(^)(ll-c/)  =  792 

25.  (y+4)(>'-5)  +  (>'-2)(>;+7)=-28 

26.  «2(:^-10)=:v2_i8ic 

27.  w(w2-ll)-7w(«+l)  =  0 

28.  ^-!+^^  =  8. 

o 

29.  If  2  rods  be  added  to  one  side  of  a  square  field  and  4  added 
to  the  other  side,  the  area  of  the  resulting  rectangular  field  will  be 
48  square  rods.  What  was  the  area  of  the  original  field?  Will 
both  roots  of  the  equation  serve  as  answers  to  the  problem?    Why? 


PRODUCTS;   FACTORING;  EQUATIONS  179 

30.  Find  two  numbers  one  of  which  is  4  times  the  other  and  whose 
product  is  196. 

31.  Find  a  number  such  that  5  times  its  square  increased  by  10 
times  itself  equals  495. 

32.  Find  three  consecutive  numbers  such  that  their  sum  is 
equal  to  three-sevenths  of  the  product  of  the  last  2. 

33.  The  difference  between  the  squares  of  two  consecutive  num- 
bers is  49.     What  are  the  numbers? 

34.  The  sum  of  the  squares  of  two  consecutive  integers  is  265. 
Find  the  numbers. 

35.  The  sum  of  the  squares  of  three  consecutive  integers  is  245. 
Find  the  numbers. 

36.  The  length  of  a  rectangle  is  1  foot  more  than  3  times  its 
width.     If  the  area  is  200  square  feet,  find  the  dimensions. 

37.  The  base  of  a  triangle  is  2  feet  more  than  twice  the  height. 
If  the  area  is  110  square  feet,  find  the  dimensions. 

38.  If  one  side  of  a  square  is  increased  by  3  and  the  other  side 
.by  1,  the  area  of  the  rectangle  formed  is  12  less  than  the  area  of 
the  square  formed  by  increasing  each  side  by  3.  Find  the  area  of 
the  original  square. 

39.  The  length  of  a  rectangle  is  7  more  than  the  width.  The 
perimeter  of  a  square  of  the  same  area  is  48.  Find  the  dimensions 
of  the  rectangle. 

40.  One  side  of  a  certain  rectangle  is  16  feet.  Its  area  is  18  square 
feet  less  than  twice  the  square  on  the  other  side.  Find  the  dimen- 
sions of  the  rectangle. 

» 

41.  A  side  of  one  square  is  6  feet  more  than  the  side  of  another 
square.  The  difference  between  their  areas  is  96  square  feet. 
Find  the  side  of  each. 

42.  An  open  box  3  inches  high  and  containing  108  cubic  inches 
is  to  be  made  from  a  square  piece  of  tin  by  cutting  out  the  comers 
and  turning  up  the  sides.  What  must  be  the  dimensions  of  the 
square  piece  of  tin? 


180 


BEGINNERS'  ALGEBRA 


In  Fig.  45  a  trapezoid  is  shown;  a  is  its  height  or  altitude,  b 
is  its  lower  base,  and  c  is  its  upper  base.  It  is  shown  in  geometry 
that    its    area    may    be   found   by    the  ^ 

formula 

area=-^-r — - 


6 
Fig.  45 


43.  Find  the  area  of  a  trapezoid  if: 

(1)  Its  altitude  is  6  feet  and  its  bases 
are  10  feet  and  15  feet. 

(2)  Its  altitude  is  9  feet  and  its  bases  are  7  feet  and  12  feet. 

(3)  Its  altitude  is  4  feet  and  its  bases  are  20  feet  and  30  feet. 

(4)  Its  altitude  is  a  feet  and  its  bases  are  -|o  and  fa  feet. 

(5)  Its  altitude  is  a  feet  and  its  bases  are  a— 2  feet  and  a+9  feet. 

44.  Find  the  dimensions  of  a  trapezoid  if  the  longer  base  is 
twice  the  shorter  base  and  the  altitude  is  5  feet  less  than  the  shorter 
base,  the  area  being  75  square  feet. 

45.  Find  the  dimensions  of  a  trapezoid  if  one  base  is  1  foot  more 
than  twice  the  other,  and  the  altitude  is  4  feet  less  than  the  shorter 
base,  the  area  being  148  square  feet. 

46.  The  bases  of  a  trapezoid  are  respectively  6  feet  and  8  feet 
longer  than  the  altitude.  Find  the  dimensions  if  the  area  is  60 
square  feet. 

47.  The  length  of  a  certain  rectangle  is  5  feet  more  than  its  width. 
Its  area  is  2^  times  the  area  of  the  square  whose  side  is  the  width 
of  the  rectangle.     Find  the  dimensions  of  the  rectangle. 


CHAPTER  VIII 

Review  and  ExTENSiori  of  Fundamental  Operations 

127.  General  view.  Let  us  stop  to  take  account  of  what 
we  have  learned  in  the  preceding  chapters.  The  most 
important  thing  that  we  have  learned  is  that  we  may  solve 
problems  by  using  letters  for  the  unknown  numbers,  stat- 
ing the  problems  as  equations,  and  solving  the  equations. 
We  found  a  number  of  different  kinds  of  equations  and 
developed  methods  for  solving  them. 

For  solving  these  equations  it  was  found  that  we  needed 
to  know  how  to  add  and  subtract,  multiply  and  divide, 
and  to  factor  algebraic  expressions.  The  examples  we  have 
worked  were  comparatively  simple,  though  they  seemed 
somewhat  difficult  because  of  their  newness  to  us.  In  the 
present  chapter  we  wish  to  gather  together  some  of  these 
algebraic  ideas,  review  them,  and  extend  them  to  more 
difficult  examples  and  problems. 

DEFINITIONS 

128.  Terms,  coefficients,  exponents.  An  algebraic  expres- 
sion may  be  made  up  of  parts  that  are  separated  from  one 
another  by  plus  or  minus  signs.  These  parts  with  the 
signs  in  front  of  them  are  called  terms.  The  expression 
Sx^-5x+2  has  three  terms:   Sx^,  -5x,  +2. 

An  expression  of  one  term  is  called  a  monomial;  one  of 
more  than  one  term  is  called  a  polynomial.  A  two-term 
expression  is  called  a  binomial;  a  three- term  expression 
is  called  a  trinomial.  Special  names  are  not  often  used  for 
polynomials  of  a  larger  number  of  terms. 

3x2  is  a  monomial. 
x^  —9  is  a  binomial. 
ic^  —  3ic  —  2  is  a  trinomial. 

181 


182  BEGINNERS'  ALGEBRA 

A  term  may  be  made  up  of  two  parts,  a  numerical  part 
and  a  literal  part.  The  numerical  part  is  called  the  numeri- 
cal coefficient  or  simply  the  coefficient. 

In  3:^^  3  is  the  coefficient  of  x"^.  The  coefficient  of  a  term  like  x^ 
is  understood  to  be  one.    The  coefficient  of  —2x  is  —2. 

The  name  coefficient  is  used  in  a  broader  sense  also.  In 
a  product  either  of  two  factors  may  be  called  the  coefficient 
of  the  other. 

In  aic2,  a  is  the  coefficient  of  x^.  In  aXj  a  is  the  coefficient  of  x; 
also  X  may  be  called  the  coefficient  of  a. 

The  choice  of  what  shall  be  taken  as  the  coefficient  depends 
upon  the  choice  of  the  letters  which  are  the  center  of  our 
interest. 

129.  Like  or  similar  terms.  Terms  are  said  to  be  Uke 
or  similar  when  they  differ  only  in  the  numerical  or  literal 
coefficients. 

3:c2  and  bx^  are  like  terms. 

bx  and  Zy  are  not. 

ax  and  hx  may  be  regarded  as  like  terms. 

130.  Degree.  The  degree  of  a  term  is  the  sum  of  the 
exponents  of  the  unknowns  that  it  contains. 

Zx"^  is  of  the  second  degree  in  x. 
ax"^  is  of  the  second  degree  in  a;. 
xy  is  of  the  second  degree  in  x  and  y. 
x'^y  is  of  the  third  degree  in  x  and  y. 

EXERCISES 

Evaluate  the  following  when  x  =  2,y  =  Z,  a  =  5,  6  =  4: 
L  x^y^  a'^x^,  by^  2.  8xy*,  ^ax^,  5a^b  3.  axy^,  ax^y,  a^xy 

4.  a*Xy  b*y,  a*xy  5.  a^x,  x^a^,  x^y^ 

The  degree  of  an  integral  expression  is  given  by  the  term 
of  highest  degree. 

3^2  —4^ -(-2  is  of  the  second  degree  in  x. 

4x3  _5^  -3  is  of  the  third  degree  in  x. 

xy  -3x  is  of  the  second  degree  in  x  and  y. 


FUNDAMENTAL  OPERATIONS  183 

THE   FUNDAMENTAL   OPEllATIONS   WITH  INTEGRAL 
EXPRESSIONS 

ADDITION 

131.  Addition  of  terms.  Rule.    To  add  like  terms,  mul- 
tiply the  sum  of  the  coefficients  by  the  common  factor: 

9xy—5xy  =  4xy 

ax-\-bx={a-\-b)x 

* 

The  addition  of  unlike  terms  can  be  indicated  only: 

ax-\-by 

EXERCISES 

Find  the  sum  of: 

1.  x\  2x,  Sx^,  5^2,   -3x,   -5,  4 

2.  xy,  Sxy,   -5x^,   -7/,  -{-x'',   -5xy,   -7x\  Sy^,  Sxy 

Put  the  following  polynomials  into  more  condensed  forms  by 
adding  like  terms: 

3.  4:m-\-Sn—27i-\-5m—7n—3m 

4.  3jc+4>;-2z-7:r+9y+llz-13>'-10z 

5.  Sx^+5x -2x^+2 -3x -5x^-5 

6.  Sab+Qa-'b-\-7ab^-2ab-3ab^+3a^b 

7.  3xY+5x^y  -^xy^  -3x^y+2xy^  -Qx^y^ 

8.  7a2+|a+fa2+3-fa+2a2 

9.  k^ -Uhi+Qkn^-\-2k^n  -3kn'^  -2n^-\-n^+3k^n-\-4:kn^ 

10.  67r-87r+97r-37r 

11.  2TrR-'!rR'+8irR-37rR+7TrR^ 

132.  Addition    of    polynomials.     How    are    polynomials 
added?     (See  Art.  61.) 

EXERCISE    I 

Add  the  polynomials; 

1.  3:^+2,  7ic-5,  2ic-4,    -3jc+7 

2.  x^-{-3x+2,  2:«2_7;3c+3,  2jc2+5:x;-8,  rt^-io,  3a:2-2a;,   -3x-f2 

3.  x^-^x^+x\  2jc2-2;c-3jc3,  3^-3x^+3 


184  BEGINNERS'  ALGEBRA 

4.  9jc+3y4-3,  7a: -23; -7,  Sx-5y+l,  Sy-2x,  7x-4. 

5.  iw'-iw+yV,  -i+w'-i«,  A+fw-f^' 

6.  ax'^-\-bx,  bx^-\-a,  ax-\-b,  ax^-{-bx+c,  bx'^-{-cx-\-a,  cx^-]-ax-{-b 

7.  x*+Sx^-2x+S,  x^-2x^-\-Sx-7,  x*-Qx-\-4,  x^-5x+2 

Expressions  in  parentheses  may  be  treated  as  a  single 
term. 

Thus,  4(a-6)+5(a-6)=9(a-6) 

for  that  is  exactly  what  is  often  intended  when  parentheses 
are  used. 

EXERCISE  II 

In  this  way  add: 

1.  Q(x+y)-5(x-]-y)+8{x+y) 

2.  4:{x+y)  -7{a-b)+8ix-\-y)  -9{a-b) 

3.  dx^-{a-b),  5x^-i{a-b),  n{a-b)-9x\  -^7x^+2{a-b) 

4.  a{x—y)+b{x—y) 

5.  7x^y-{x-y),  -9x'^y+9{x-y),Sx^y-5{x-y),  -Qx^y-\-4:{x -y) 

6.  2x^y^-(x^-y^),        -12x^y^-7{x^-y^),       +15x^y^+5{x^-y^), 

7.  (x  -y)  -9xy\  -Qix  -y)-\-7xy^,  4(:»  -y)  -2xy\  -S{x  -y)  -5xy' 

8.  Add  4:a-\-2b-\-Sc,  6a+36+2c,  5a+7b+9c 

What  does  each  expression  and  their  sum  become  when  a=  100, 
J=10,  c=l? 

9.  Add  5x-2y-{-z,  8x+3>'-4z,  Qx-2y-3z 

Evaluate  each  expression  and  the  sum  when  jc  =  100,  y=lO,  z=l. 

10.  Add  8:*: -23;,  15:c-7>;,  9x+Sy,  12x-y,  Sx+9y 

If  x=12y,  find  the  value  of  each  and  the  value  of  the  sum  in 
terms  of  y. 

11.  Add  Sx+2y-z,  4x-y-\-9z,  Qx+y-lOz,  5:«-23;+llz 

Find  the  value  of  each  expression  and  the  sum  in  terms  of  z  when 
a:  =  3z  and  y  =  2z. 


FUNDAMENTAL  OPERATIONS  185 

MULTIPLICATION 

133.  Products  of  monomials.  Rule.  Multiply  the 
numerical  coefficients  and  affix  to  each  letter  an  exponent 
which  is  the  sum  of  the  exponents  of  that  letter  in  the 
monomials  to  be  multiplied. 

State  the  rule  of  signs.     (See  Art.  67.) 

EXERCISES 

1.  Sx  •  5x  2.  5x  '  2x^ 

3.  3^3 .  ^ax  4.  -2x^  '  Sax 

5.  a  •  Sab  6.  Sx^  •  —  7xy 

7.  ab'c  •  a^-hc  8.  21^26  •  -Sa^b 

9.  3w2  .  2/>2  .  4«2  10.  3^  •  4/»  •  ts""  •  2/^^ 

11.   -9x^y  •  -8.r'y  12.  -3ay  •  -^x'^y  •  2xy^ 

13.  ix^y  •  ^xy^  •  -f^c^y  14.  ^r^^i  •  -j^rsH^ 

15.  3:r  •  ixy  '  -5y  16.  fax  ' -j^by  '  -2xy 

134.  MultipUcation  of  poljmomials.  The  product  of  two 
polynomials  is  the  sum  of  the  products  of  each  term  of  the 
one  with  each  term  of  the  other. 

As  was  indicated  in  Art.  Ill,  it  is  generally  most  conven- 
ient to  arrange  the  multiplication  as  in  arithmetic  with 
similar  terms  under  similar  terms  in  the  partial  products: 

x^+Sx-5 
2x-S 


-3^2_9^_l_15 

2^+3ic2-19;t;+15 

Check  by  substituting  a  value  for  x,  say  3,  in  the  expres- 
sions to  be  multiplied  and  in  the  product,  then: 

9+9-5  13 

6-3  3 


54+27-57+15  39 

13 


186  BEGINNERS*  ALGEBRA 

EXERCISE  I 

Simpler  exercises.    Find  the  products: 

1.  x{x-l)  2.   -x{x-2) 

3.  7r(a-4)  4.  x{x^+3) 
5.  2Trr{r-h)  6.  (n+2)(n-5) 

7.  (3w+l)(2w-3)  8.  (x^-7x-]-2){x-S) 

9.  (:«2-5:»:+3)(ic2+2:K-3)  10.  (3;z2-2w+4)(4»2-f.;i_3) 

11.  (2fl2_3a+l)(2a2+3a-l)  12.  (:c2_|_3^_5)(^2_^^2) 

13.  (2/2-3/+5)(2^-3)  14.  {x^-x^-x-\-l)ix^-2x-\-2) 

Arrange  the  following  in  the  order  of  the  descending  powers  of  x, 
the  highest  power  first  and  so  on : 

15.  x^+x^-l-^2x  16.  4+x3-2x-x2 

17.  3(3c2+3a;+l)+2(;^;-3)-5     18.  S(x-l)Xx-\-2) -2{x+2)x 

19.  {x^-x){x-2)-ix-3){x-2) 

20.  ix+Sy-2{x-{-dy-\:{x-\-S)-5 

In  multiplying  two  polynomials  it  is  desirable,  though  not  abso- 
lutely necessary,  to  arrange  them  in  some  regular  order ;  for  instance, 
according  to  decreasing  or  increasing  exponents.    Why? 

21.  (3/-2/2_|_5)(^_3/) 

22.  ih^-2h-\-h'+2)i2+Sh^-h) 

23.  (2+x\r-x-2x'^){x^-l-x) 

When  terms  of  the  regular  order  are  missing,  it  is  often  well  to 
leave  room  for  them  when  setting  down  the  work. 

24.  (h^-Sh+2)(2h^-S) 

25.  (4ic3-3x-l)(2:K3_j_2x2-2) 

26.  {x-\-x^-2)(Sx^+l) 

Find  the  shortest  way  of  multiplying  Exercises  27  to  33: 

27.  n{n+2){n+S) 

28.  (w-l)(n+2)(w-3) 

.  29.  {p-l){p+l){p'-\-l) 

30.  {x-2){x-2){x-\-2){x-\-2) 

31.  {x-y){x^-xy+y^){x-\-y) 


FUNDAMENTAL  OPERATIONS  187 

32.  (o-2)((i2-f4)(a+2) 

33.  (3x-l)(3^-l)(3x-l) 

EXERCISE   II       , 

More  difficult  exercises: 

1.  iy'-\-2y^-^y-l){y'-y+l) 

2.  (n3+3w2-2»+l)(w2-2»+l) 

3.  {x^-2xy-\-y^){x'^-{-2xy-^y^) 

4.  (4ft -i) (5^+1) 

5.  (3x4+6x33; -9xy34-12/)(2a;2 _3^y_|.y^ 

6.  (l+2«+3w2+4»3+5w4)(2 -3»+4«2 -5»«) 

7.  (3x-2x3+2x*-7x2+5)(3+3x2+2a:) 

8.  (x2-4xy-2>'2)(x2+4.T>;-23;2) 

9.  (3c2-3xy-5>'2)(:r2-3xy+5/) 

10.  {x^-2x^y-]-xy^){x''+2xy-y^) 

11.  (jc4+6x2y->;2)(x4-|-6x2y-43'2) 

12.  (a+x2)(a2_2ax2+x4) 

13.  (ax2+4x3)(a2-ax+a;2) 

14.  (3x*-5)(3x2+5)(3x2+5)(3x2-5) 

15.  (x-l)(x^+l){x+l)(x^+l) 

16.  (2x-l)(4x2+l)(2x+l) 

17.  ia''+ab+b^){a'-ab-{-b^) 

18.  (3x3y  -2x44-5x2>;2 +3,4) (^2  _xy-{-2y^) 

19.  (5x*-7x33;-2x>'3-y)(x2-2x>'-3;2) 

20.  (x'-3x2>f+3x>'2-y3)(x2-y2) 

21.  (x^-Axy-5y^)(x^-y^-xy) 

22.  (a-6)(6-a)(c-a) 

23.  {a^+¥+c^-bc-ca-ab)(a+b+c) 

24.  (6+c)(&-c)-h(c+a)(c-a)  +  (a+6)(o-6) 

25.  (x+y)(x4-x3>'+x23;2_jcy3+>;*) 

26.  {x-y)ix-\-2)(x-S){x+^) 

27.  (x+l)(x2-2)  +  (x2-l)(x+3) 

28.  {a-b){b-c){c-a) 


1S8  BEGINNERS'  ALGEBRA 

SUBTRACTION 

135.  Subtraction  of  poljmomials.     How  is  one  polynomial 
subtracted  from  another?     See  Art.  64. 

EXERCISES 

1.  Subtract  Sn  -2  from  5«+7. 

2.  Subtract  4w2  -2w+3  from  7n^  -5»+l. 

3.  From  -7x^y^+13x^y+l5x*y  take  4xY+7x^y  Sx^. 

4.  From  5k*  -3k^  -2^24-5ife+2  take  2k^  -5k^-\-2k^  -3k  -5. 

5.  From  Sx^y  -2xy^  -x^  take  Ox^y  -Qxy^+Sx\ 

6.  From  2a^b -6a^b^-{-7ab^-b*  ta^.Ta^ft+Sa^fiJ -3a6' -2b*. 

7.  Take  aac  from  bx. 

8.  Subtract  a;c— &y  from  bx-i-ay. 

9.  Subtract  3a3jc-8a2a;2+9ajc8-2jc4from7a3x-9a2x2-10aA;»-3a:< 

10.  Subtract  S{x-\-y)  from  -7(x+>')- 

11.  Subtract  2x*-^Q{x -y)  from  3^4 _(-,; _y). 

12.  Subtract  3(jc  -y)  -7(o+6)  from  2(:»;-y)+3(a+6). 

13.  Fromx^-(a'-b)  +  (x-y)  take  3x2-2(fl-fc)+4(x-y). 

14.  From  aac  — (y— z)  take  — 3(y— z). 

15.  Subtract  i»ic-2(a-&)  from  c;c+2(a-6). 

16.  Subtract  3ic2-7a:+l  from  2;c«-5x-3,  subtract  that  differ- 
ence from  zero,  and  add  this  result  to  2^^  —  S:*;— 4. 

17.  ix»-x^+x+l)-(x^+x^-x+l)  =  ? 
Reduce  to  simpler  form: 

18.  ( -3a»+563  -c»)  - (a^+fta  -c^  -Sabc) 

19.  (2uv  -»«  -3m«)  -  (w'+v*  -2m») 

20.  iSxy+2x^)-'(^x*-2xy^+3yx) 

21.  3x  -2(2x2  _3y)  _3(a:4-2:r«  -y) 

22.  a;«-(x4-5)(:x;-l) 

23.  A3- (A+ 1)2+2(^-2) -5 

24.  (w-2)-2(w-2)-(3n-6)+5 


FUNDAMENTAL  OPERATIONS  189 

136.  Parentheses.  Parentheses  are  used  for  grouping 
terms  that  are  to  be  treated  alike.  They  are  removed 
from  the  work  when  the  indicated  operations  are  worked  out. 
(See  Art.  18.) 

EXERCISES 

Perform  the  operations  indicated: 

1.  (x-S){x-4)  2.  :c-3(a:-4)  3.  {x-S)x-4: 

4:.  xiy-2)-y{x-S)        o.  x{y-z) -a{b-c) 

137.  Parentheses  within  parentheses.  Brackets  [],  braces 
{ },  and  the  bar,  a  —  b,  are  often  used  for  the  same  purpose 
as  parentheses,  especially  to  avoid  any  confusion  when 
one  set  of  parentheses  incloses  another. 

5  (r  —  (n  — 1)3)  means  that  n  —  1  is  to  be  multiplied  by  3,  the 
result  subtracted  from  r,  and  that  result  multiplied  by  5. 
When  such  expressions  are  written  with  brackets,  there  is 
less  danger  of  confusing  the  pairs  of  signs : 

5[r-(w-l)3] 

In  such  cases  we  may  best  simplify  the  expression  by 
performing  the  indicated  operations,  beginning  with  the 
inner  parenthesis: 

5  [r-(n-l)3]  =  5  [r-(3w-3)] 
=  5  [r-3w+3] 
=  5r-15w+15 

EXERCISES 

Perform  operations  indicated: 

1.  o-(+w-[3/+5]) 

2.  S+[{x-2)  +  {x+2)\ 

3.  S-[ix-2)-{x+2)] 

4.  x[{x-S)^-4.x] 

5.  x^-{x-y)-[x-{Sx+4:y)] 

6.  x-ix'-2)+[ix-S)-i2x+5)] 

7.  da^-[a{a-b)-b{a-Sb)+ab] 


190  BEGINNERS'  ALGEBRA 

138.  Insertion  of  parentheses.  It  is  often  important  to 
put  certain  terms  of  an  expression  into  parentheses.  It  will 
be  noticed  that  when  parentheses  with  a  plus  sign  in  front 
are  removed  the  signs  of  the  terms  taken  out  remain  un- 
changed. So  in  the  inverse  operation  of  putting  the  term 
back  in  parentheses  the  signs  of  the  terms  are  not  to  be 
changed : 

a+M  — 3  =  a+(w  — 3) 
On  the  other  hand,  if  the  sign  before  the  parentheses  to 
be  removed  was  nainus,  the  sign  of  every  term  taken  out 
was  changed.      So  also  in  putting  the   terms   back  into 
parentheses  with  a  minus  sign  in  front. 
Since  a— (w+3)  =a— n  — 3 

a— w  — 3  =  a  — (n+3) 

EXERCISES 

Put  the  last  two  terms  into  parentheses  in  Exercises  1  to  8: 
1.  a-\-b-\-c  2.  a+b-c  3.  a-b-\-c 

4.  a-b-c  5.  Zx-2a-^         6.  {a-\-by-a-b 

7.  {a-by-a-\-b         8.  x'^-a'^-x+a     9.  x^-x^-x-l 

In  Exercises  10  to  18  put  first  two  terms  in  one  parenthesis  and 
last  two  in  another": 

10.  a-\-b-c-d  11.  a-\-b-{-c-\-d  12.  a-b-\-c-d 

13.  a-b-c-\-d  14.  Zx^ -2-{-^x^ -^        15.  x-'-2-2x-'-\-^ 

16.  x'^-2x-\-a-b        17.  x^-}-^x-c-d  18.  x^^-x^-x-l 

Group  in  pairs: 

19.  x^-l+2x^-4^-Sx-\-S        20.  a-b-c-d+e-f 

Group  in  threes: 

21.  x^-3x+2-2y^-\-Gy-4:       22.  x^+5x-S-2y^-7y+5 

DIVISION 

139.  Exact  division  by  a  monomial.  To  multiply  one 
number  by  another  we  introduce  the  second  as  a  factor. 


FUNDAMENTAL  OPERATIONS  191. 

To  multiply  3a  by  x,  introduce  the  factor  %;  the  result  is 
2tax. 

The  inverse  operation  of  dividing  a  product  by  one  of 
its  factors  is  accompHshed  by  rejecting  that  factor. 

Divide  3ax  by  a;  the  result  is  Zet>  or  Zx 

Divide  63  by  7;  the  result  is  9  •  7  or  9. 
Zx^'6=Zx  =  x 

So  also  a'ra=\' £t-T0=l 

In  all  of  these  cases  the  quotient  is  the  rest  of  the  term 
with  the  division  factor  omitted. 

EXERCISES 

Divide: 

1.  ab  by  a  2.  xy  by  y  3.  dbc  by  ah 

4.  ahc  by  ahc  5.  12a  by  3  6.  VJxy  by  x 

7.  365/  by  I2t  8.  lO:^;*  by  x  9.  ax-^hx  by  x 

10.  ab^ac-\-ad  by  a  11.  (2.x3  _4jc2+6^)  by  2jc 

140.  Use  of  exponents.     In  multipl3ang  and  dividing  ex- 
pressions what  is  done  with  the  exponents  ?    (See  Arts.  97, 98.) 

141.  A  peculiar  form:  jt°.     In  applying  the  exponent  rule 
to  such  a  case  as 

o^-^x^ 
we  have  x^-^x^=^ x^~^ = x^ 

What  does  x^  mean?    We  will  give  it  a  meaning  consistent 
with  some  of  our  other  work. 

If  we  divide  :r^  by  ::c^ 

by  discarding  the  factors  we  have 

To  make  the  rules  consistent,  we  must  say  that  x^  means  1 . 
Note  that  ic  is  ic^  not  x^. 


192  BEGINNERS'  ALGEBRA 

142.  Division  of  monomials  by  monomials.     Make  your 
own  rule.     (See  Art.  98.) 

EXERCISES 

1.  Divide  a'  by  a«  2.  Divide  5»'  by  n^ 

3.  Divide  iirR^  by  ^R  4.  Divide  35^33;5  by  Tac^y' 

5.  Divide  2^  by  2^  6.  25^  ^252 

7.  3Qax»-i-12x^  8.  Qx*^-2x 

9.  -Sx^y-^xy  10.  12x^y>-^x^y 

11.  10x2y3^_5icy2  12.    -12a':x:2y-^-6a2ncy 

13.  15a^bx^y^^5aH^  14.  25(1^63^2  ^5^3^,^ 

15.  Divide  39^^^?  by  -Sm^n  16.  Divide  -^9x^y  by  +7ify 

17.  Divide  -Qa^bc  by  -2ab  18.  28a2Jc2y-^7x2y 

143.  Division  of  a  polynomial  by  a  monomial.     Rule. 
Divide  each  term  of  the  poljmomial  by  the  monomial. 

EXERCISES 

1.  (Qx*-4x)-i-2x  2.  {2ix^-3Qx*-20x^)^^x^ 

3.  {3Qx''y*^42xy)-T-Qxy^         4.  (9:^4  - 12:^3)  ^ -3^2 

5.  {10x^y^-15x^y^+5x^y*)^5x^y 

6.  iS2k^n*-lQk*n^)-i-Skn^ 

7.  (a2a;3-a«:x;2)^^2^  g,  (a^ic^-a^jc') -^ajc» 

9.  {aH^-a^x^-ax*)-^ax^         10.  (4a2i>-6a'6»+12o«63)  ^2a6 

11.  il0a^x*-15aH*-\-20aH')  -^  -5a^x^ 

12.  (6w»w2+12«2w3-18ww<)-^6ww2 

13.  4a(:x;-5)^(3c-5)  14.  (l-3c)-^(:*:-l) 
15.  3a(a;-l)^(j«:-l)  16.  (ic-2)2-(a;-2) 
17.  36o(a;  -y)  ^9(y  -a:)  18.  36a(:c  -y)  -i-^a 

19.  36a(:«;-y)-^6(ic-y)  20.  5(:«+l)(«-3)  ^5(:x;-3) 

21.  [(:,_y)2-2(«:-y)]H-(:.-y) 

22.  l^xia  -b)  -2(a  -6)2]  -  (a  -6) 

23.  [7x{a-b)-8a{a'-b)]^{a-b) 

24.  (:^2-l)^(«;-l)  25.  (a:»-2:r+l) -(*-!) 
26.  (a:2-3a:-4)-i-(x+l)  27.  (a;2-4) -T-(:r+2) 


FUNDAMENTAL  OPERATIONS  193 

28.  {x^-3x-^2)-^ix-2)  29.  3(:r2-6;r+9)^(af-3) 

30.  {x^-^10x-^25)^{x+5) 
Find  exact  divisor  for: 

31.  x^-25  32.  ^2-8:«;+16  33.  a;2-2x-15 
34.  a^-lOO                   35.  3;2_iO;y+21  36.  x^-A9 

37.  ^2-81  38.  2a2+19a+9  39.  4:^2-25 

144.  Division  of  a  polynomial  by  a  poljmomial.     The 

following  example  from  arithmetic  will  show  how  to  proceed 
when  it  is  desired  to  divide  one  polynomial  by  another. 
Divide  8673  by  21: 

413,  quotient 

21|  8673 
84_ 

27 
21 

63 
63 

This  may  be  put  in  a  fuller  form: 

400+10+3,  quotient 
2Q+1|  8000+600+70+3 
8000+400 

200+70 
200+10 

60+3 
60+3 

If  we  represent  10  by  t,  the  work  takes  this  form: 

4/2+  t+S,  quotient 
2/+l|8/^+6/2+7/+3 
8/^+4/2 

2/2+7^ 
2t^+t 

6/+3 

6/+3 


194  BEGINNERS'  ALGEBRA 

Apply  to  {x^-Sx+2)^{x-2) 

x—1,  quotient 
x-2\x^-dx+2 
x^-2x 

-  x+2  ^  .     , 

-  x-^2 

EXERCISES 

Divide: 

1.  x^+9x^+2Qx+24:  by  x+2 

2.  x^-5x^+x+10  by  x-2 
S.  x^-Qx^-\-5x+12hy  x-4: 

4.  x^-Sx^-4:X+12hyx-\-2 

5.  2n^-n^-\-4:n'^+^n -Shy  n^-n+S 

6.  4+12a+13(i2+6a3+(z4by2+3a+a2 

In  the  multiplication  of  polynomials  the  arrangement  of 
terms  in  ascending  or  descending  powers  of  some  letter  is 
merely  a  matter  of  convenience.  But  in  division  such  an 
arrangement  is  necessary.  It  is  desirable  to  keep  like  terms 
under  each  other.  This  may  be  accomplished  by  means  of 
a  vacant  place  when  there  is  a  missing  term.  In  arithmetic 
such  vacant  places  are  filled  by  means  of  a  zero. 

Divide: 

7.  a;3-40+233C-8ic2  by  af-5 

8.  4w3  -  Um-\-2m^+S  by  4m+2w2  - 1 

9.  x*-10x^-\-15x-Qhy  x-'-^-Sx-^S 

10.  x*-x^-\-2x^-x-^lhy  x^-x-\-l 

11.  a;4_5;;c8+i5:x;2-23jc+20  by  ic2-3^+4 

12.  5k^-S-5k+Qk^-Sk^hy  Sk^-2k-l 

SUPPLEMENTARY   EXERCISES 

1.  (6a2-10+o-4o3+a*)-^  (5-3aH-a«) 

2.  (3a-8a2+9+a*-a3)-h  (a2-3-2a) 

3.  (&*-4a»^>^-a262^-a4^-4a^>'»)-^  (a^-^-ab) 

4.  (6A'^-9/f*+10/f'-18A2-5)-^(3A«+5) 


FUNDAMENTAL  OPERATIONS  195 

5.  ix*-\-x^-\-y^)  ^  {y^  -xy+x^) 

6.  (a*-l)^  (a+1) 

7.  (x4-12x3+36ic2-25)H-(a;2-6x+5) 

8.  {4:X*-lSx^y^+9y*)-i-i2x^-^xy-Sy^) 

9.  (x9-a9)-^(^+ic»a»+a«) 

10.  (ici2-^)i2)^  (a;4-6<) 

11.  (9a:*  -49:^2^2+16/)  -^  (S.r^  -Sjc^'  -43/2) 

12.  (8a«+l)-^  (2^2+1) 

13.  {a^-\-2b^-2a*b-5a¥-\-a^b^+a^b>)-^  {a^+2b^-ab) 

14.  (a2-4a6+462_c2)-(a-26+c) 

145.  Inexact  division.  If  the  division  is  inexact,  that  is, 
if  a  remainder  is  left  after  aU  the  dividend  is  used,  proceed 
in  the  same  way.  The  division  should  be  carried  on  until 
the  highest  exponent  of  the  remainder  is  less  than  the  highest 
exponent  of  the  divisor. 

Illustration  1.  a;— 5+,  quotient 

x+2\x^-3x-\-2 
x^-^2x 


-bx^-2 

-5a; -10 

+  12,  remainder 

Ilk 

istrat 

ion  2. 

x  —  \-\-,  quotient 

^+3| 

a;2+2:c-5 
x-'-^Zx 

-x-b 
-x-Z 

—2,  remainder 

EXERCISES 

Divide: 

1. 

x^- 

x2-3by  3c2 

2. 

2y3. 

-2>'2-3;y+lbyy 

3. 

3y- 

_  53,3  _  23,2- 

y_|_lbyy» 

1%  BEGINNERS*  ALGEBRA 

4.  x+1  by  x—l 

5.  a:2-j-lby  «+l 

6.  a;2+«;-25by«;-3 

7.  ^'-lU2_j_2iby^-5 

8.  2:x:2-5jc+7bya:+3 

9.  x^-\-5x^-h7x~3hy  x^+Sx-i-l 

10.  (4:r'+18-9x2_i5^)^(^2_4;^_^3) 

11.  (20:^3 -5x2 -4:x:+7)^  (5^2  _i) 

12.  {6x^-13x^+9x-2)-~{2x^-Sx+l) 

146.  Division  as  a  fraction.  Instead  of  carr3nng  out  the 
division  of  one  polynomial  by  another  as  in  the  preceding 
articles,  it  is  the  usual  practice  in  mathematics  to  express 
the  division  in  the  fractional  form  and  to  treat  the  problem 
from  that  point  of  view.  This  method  will  be  considered 
in  a  later  chapter.  It  is,  however,  sometimes  very  impor- 
tant that  the  long  division  be  actually  performed. 

SOLUTION  OF  EQUATIONS 

147.  Linear  equations  in  one  unknown.  An  equation  of 
the  first  degree  in  one  unknown  is  called  a  Hnear  equation. 
State  the  rule  for  solving  a  linear  equation.  Why  should 
the  solution  be  checked? 

EXERCISES 

Solve  and  check: 

1.  3x-'2-\-7x=Sx-5 

2.  50-10x-3+2«  =  3 

3.  2«+3  =  16-(2w-3) 

4.  8(/-3)-(6-2/)  =  2(/+2)-5(5--/) 
6.  7(25-jr)  =  2x+2(3x-25) 

6.  15{x-l)=2{7+x)-4.(x-\-S) 

7.  ax=  -3jc+& 

5.  ai==c{t-{-h).    Solve  for/. 
9.  .5«+2. 75  =  3.25^-1 


FUNDAMENTAL  OPERATIONS 


197 


10.  2{x-a)+3ix-2a'j=2a 

11.  S{x-^d+b)-\-2(x+a-b)  =  Qb 

12.  {m+n)t+in-m)i  =  n\     Solve  for /. 


13.  35w(5-2)-7(w-3)  =  l7 

15-  2 — ^=^^ 

17.  10-^=10+^'  • 

21.  x{a-{-b)  =  a^-¥ 

23.  ax+a  =  a3-:<:+a2-l 

3x— a     2x—a 
25.  — ^ —  =  — 7i — 


14.  8l/-(3-5/)-h7]  =  5/-ll 
2       2 


^«-|-¥ 


x-3     x-{'2     2     T_7 
18.      2        "^-3"^    3 

20.  x(a-b)^a^-b^ 

22.  ax-bx=a^+2a-ab-2b 
a(x—a)     x-^2 


24. 


26. 


4 
5x  — a 


~    2 

2:c-a 


148.  Factorable    equations    in    one    unknown.     If    an 

equation  of  a  higher  degree  than  the  first  can  be  put  in 
the  factored  form,  it  can  be  easily  solved.  What  is  meant 
by  the  factored  form  ?  State  the  methods  of  solving  such 
equations. 

An  equation  of  the  second  degree  is  called  a  quadratic 
equation.  With  your  present  knowledge  can  you  solve  any 
quadratic  equation  given  you? 

How  many  roots  does  an  equation  of  the  first  degree  have? 
How  many  roots  does  an  equation  of  the  second  degree  have? 


EXERCISES 


Solve  and  check: 

1.  :«:2-7:x:  =  0 

3.  «2+12  =  7w 

5.  225  =<r* 

7.  2.c2-5:c  =  3 

9.  {a-\-b)x^=^cfi+a*b 
11.  9a;2-16(1-jc2)=0 


2.  6:c+91=x« 
4.  3-2:c-a;2  =  0 
6.  3jc24-8a;+4  =  0 
8.  x^-2ax+a^=0 
10.  Sx^-2ax-bx  =  0 
12.  9x2=13-4x 


198 


BEGINNERS'  ALGEBRA 


13.  {x-7){x-5)=0  14.  {x-2){x-S)  =  ix-2)S 

15.  (a;-3)(x-4)-(2x-3)(x-7)  =  12 
,^    x-3    x^-4:     x-o 

^^'  ~2 5~^T~ 

17.  (a;-3)(2:r+l)-l  =  4^-(3:x;-l)(2:x;+5)+3 

^^'  ~^r~  7  r  5 

19.  {2x-\-a){x-^a)  -2a{x-\-2a)  =  0 

x{x-\-a)     W     x^-2a^ 
5  ~ 

22.  {x-2Y-{-'S{x-2)  =  0 

24.  7{x-3)  =  (x-S)2x 


2^-2  5  5 

21.  {x-l)^-\-{x-l)=0 

23.  x{x-S)-5{x-S)  =  0 


149.  A  set  of  linear  equations  in  two  unknowns.  State 
rules  for  solving  a  set  of  linear  equations  in  two  unknowns 
and  show  under  what  circumstances  each  method  is  the 
most  desirable.  How  many  solutions  has  such  a  set  ?  What 
kind  of  a  figure  is  the  graph  of  a  linear  equation? 


EXERCISES 


Solve  and  check.     Draw  graphs  for  the  first  three. 


1.  5-p-^2q=0 
7-5/>+g=0 

3.  y=17jc-31 
15a:+3>'  =  39 

5.  Solve  for  v  and  u: 

v—u=a 
v+u=e 

7.  P-{-Q=n2 
5P=9Q 

9.  %x+y=l 


2.  2:»;+73;=^38 
3:c+4y=31 

4.  3:^=15 +5>' 
2:t;+>'  =  205 

6.  Find  a  and  b  in  terms  of 
x  and  y: 
x=3a+2b 
y  =  a—bb 

x  —  \b  =  y 

10.  7a;+4  =  4y-3;c+3 
bx 


y+2y=l| 


11. 

l^h' 

f-.=  16 

13. 

.+%-'=8 

^^-.=  13 

15. 

f+7„  =  2 

-+^^=? 

17. 

/-I     j-2 
2         3 

FUNDAMENTAL  OPERATIONS  199 


12   ^-^-2 
12.    7     2"^ 

2ic-|=-12 
5 

14.  ba-nh  =  ^ 
6a-^°^^  =  2 


•v-1 

ic+5      . 


18. 


a:+6     y+7^2 
5  6       5 

5  =  2/  x+>'  =  0 


160.  The  stating  of  problems.  The  problem  to  be  solved 
must  be  stated  in  algebraic  language.  Definite  rules  that 
are  sure  to  lead  to  the  correct  statement  cannot  be  laid 
down.  But  certain  ways  of  attacking  problems  may  be 
suggested  as  affording  some  help.  Two  things  are  absolutely 
essential  to  success  in  stating  problems: 

Firsty  a  clear  understanding  of  the  relations  between  the 
quantities  involved  in  the  problem. 

For  instance,  in  rate  problems  such  as  involve  walking, 
rowing,  etc.,  it  is  necessary  to  know  the  fundamental  relation 
distance  =  rate  X  time 
d-=rt 
In  digit  problems  the  law  underlying  the  Arabic  notation 
must  be  known : 

273  =  2X100  +  7X10  +  3 
Second,   famiHarity  with   the  ways   in  which   ideas  are 
expressed  in  algebraic  symbols. 

For  instance,  the  square  of  the  sum  of  two  nimibers. 


200  BEGINNERS'  ALGEBRA 

The  area  of  a  rectangle  when  one  side  is  3  more  than  the 
other : 

x(x-\-d)  or  x{x-S) 

Furthermore,  in  some  cases  quite  a  little  ingenuity  is 
required  to  detect  the  relations  that  will  be  of  use  in  the 
given  situation. 

With  this  equipment  one  is  ready  to  attack  a  given  prob- 
lem. The  following  four  steps  form  a  satisfactory  plan  of 
attack.     (See  Art.  36.) 

(a)  Determine  the  nature  of  the  problem.  Is  it  a  rate 
problem,  a  digit  problem,  a  triangle  problem?  Recall  the 
fundamental  relations  involved  in  such  problems. 

(6)  Look  for  some  equality  that  is  impHed  in  the  prob- 
lem, and  state  this  equality  in  words.  Several  equalities 
may  be  suggested  by  the  problem,  but  choose  one  that 
seems  satisfactory, 

(c)  Select  some  one  of  the  unknown  quantities  of  the 
problem  to  be  the  tmknown  of  the  equation. 

(d)  Translate  this  equality  in  terms  of  the  unknown  and 
the  known  quantities  of  the  problem. 

Show  how  this  method  has  been  followed  in  the  illustration^ 
on  pages  46,  47,  and  48. 

Let  us  apply  the  plan  to  the  following  problem: 

An  automobile  running  at  the  rate  of  25  miles  an  hoiir 
made  the  distance  between  two  towns  in  2  hours  less  than 
a  second  automobile  running  15  miles  an  hour.  How  long 
did  it  take  the  first  automobile  to  make  the  trip? 

a)  This  is  a  speed  problem  (see  Art.  42).  It  has  to  do 
with  what  we  call  uniform  motion.  If  a  man  travels  at  the 
rate  of  r  miles  an  hour  for  t  hotirs,  he  travels  a  distance 
of  rt  miles. 

distance  =  rate  X  time 


FUNDAMENTAL  OPERATIONS  201 

b)  The  distances  in  the  problem  are  equal. 

Distance  first      ]       j  Distance  second 
automobile  runs  J        (automobile  runs 

c)  The  time  is  unknown.     It  is  convenient  to  arrange  the 
various  quantities  of  the  problem  in  a  table : 


Rate 

Time 

Distance 

First  automobile 

Second  automobile .... 

25 
15 

t 
/+2 

25/ 
15(/+2) 

d)  The  equality  becomes 

25/ =  15(^+2) 


REVIEW    PROBLEMS 

1.  Divide  $270  among  A  and  B  and  C  so  that  B  may  have  $25 
more  than  A,  and  C  may  have  $10  more  than  B. 

2.  Three  men  entered  into  a  contract  to  divide  the  profits  of  a 
certain  business  deal  in  the  following  way :  A  was  to  receive  twice 
as  much  as  B  and  a  bonus  of  $10 ;  C  was  to  receive  3  times  as  much 
as  B  and  a  bonus  of  $20.  If  the  profits  were  $1041 .66,  what  was 
the  share  of  each? 

3.  A  sum  of  money  s  is  to  be  distributed  among  3  men.  B  is  to 
have  twice  as  much  as  A,  and  C  is  to  have  ^  as  much  as  B.  How 
much  did  each  receive? 

4.  The  sum  of  2  numbers  is  35.  Their  difference  is  27.  What 
are  the  numbers? 

5.  The  sum  of  2  numbers  equals  twice  their  difference.  The 
sum  is  also  2  more  than  3  times  the  difference.  What  are  the 
numbers? 

6.  The  difference  between  two  numbers  is  6.  The  sum  of  their 
squares  is  50.    What  are  the  numbers? 

7.  Find  3  consecutive  integers  whose  sura  is  69. 

8.  Find  4  consecutive  odd  integers  whose  sum  is  32. 

14 


202  BEGINNERS'  ALGEBRA 

9.  The  sum  of  2  consecutive  even  integers  equals  the  product 
of  the  lesser  one  and  the  odd  integer  between  them.  What  are  the 
2  even  integers? 

10.  Of  the  3  angles  of  a  triangle  the  first  is  twice  the  second  and 
the  third  is  20°  more  than  the  second.  Find  the  number  of  degrees 
in  each  angle. 

11.  I  have  $3 .  30  in  dimes,  nickels,  and  quarters,  ha\dng  34  pieces 
of  money  in  all.  The  number  of  dimes  is  2  more  than  the  number 
of  quarters.     How  many  pieces  have  I  of  each  kind? 

12.  A  certain  nmnbcr  has  3  digits.  The  tens  digit  is  4  more 
than  the  hundreds  digit  and  1  more  than  the  units  digit.  The 
number  itself  is  5  more  than  20  times  the  sum  of  the  digits.  Find 
the  number. 

13.  A  certain  number  has  2  digits.  The  sum  of  the  digits  is  8. 
If  18  is  subtracted  from  the  number,  the  digits  will  be  reversed. 
Find  the  number. 

14.  A  boy  riding  a  bicycle  at  the  rate  of  12  miles  an  hour 
started  to  overtake  a  man  who  had  left  the  same  place  4  hours 
earlier  walking  at  the  rate  of  3  miles  an  hour.  How  long  was  it 
before  the  boy  overtook  the  man? 

15.  A  passenger  train  leaves  New  York  at  the  rate  of  45  miles 
an  hour.  Three  hours  earHer,  a  freight  left  New  York  traveling 
in  the  same  direction  at  the  rate  of  24  miles  an  hour.  When  will 
the  passenger  train  overtake  the  freight? 

16.  Two  airplanes  fly  over  the  same  course,  one  in  5  hours,  the 
other  in  3  hours.  One  can  fly  40  miles  an  hour  more  than  the 
other.  At  what  rate  does  each  fly  over  the  course?  What  is  the 
length  of  the  course? 

17.  A  and  B  start  from  the  same  place  at  the  same  time  and 
travel  in  the  same  direction,  A  at  the  rate  of  15  miles  an  hour  and 
B  at  the  rate  of  25  miles  an  hour.  How  far  apart  are  they  in  2 
hours?     In  /  hours?    When  will  they  be  50  miles  apart? 

18.  A  and  B  of  the  last  exercise  are  in  airplanes.  A  flies  5  hours 
and  is  compelled  to  stop.  B  flies  7  hours  at  a  rate  which  is  twice 
that  of  A  and  lands  at  a  point  450  miles  beyond  where  A  lands. 
At  what  rates  do  they  fly? 


FUNDAMENTAL  OPERATIONS  203 

19.  If  A  and  B  start  from  the  same  place  at  the  same  time,  but 
go  in  opposite  directions,  A  at  the  rate  of  10  miles  an  hour  and 
B  at  the  rate  cf  8  miles  an  hour,  how  far  apart  will  they  be  in  2 
hours?  In  /  hours?  How  long  will  it  be  before  they  are  90  miles 
apart? 

20.  A  and  B  start  at  the  same  time  from  places  100  miles  apart 
and  travel  toward  each  other  at  the  same  rate.  They  meet  in 
2  hours.     At  what  rate  do  they  travel? 

21.  A  and  B  start  at  the  same  time  from  places  100  miles  apart, 
A  at  the  rate  of  15  miles  an  hour  and  B  at  the  rate  of  25  miles  an 
hour.     When  will  they  meet? 

22.  A  and  B  started  at  the  same  time  from  towns  140  miles  apart 
and  traveled  toward  each  other  at  the  same  rate.  B  was  delayed 
for  1  hour  by  engine  trouble.  They  met  in  4  hours.  What  was 
their  rate  of  travel. and  where  did  they  meet? 

23.  A  and  B  started  at  the  same  time  from  two  towns  300  miles 
apart  and  traveled  toward  each  other.  A  could  travel  5  miles  an 
hour  more  than  B.  They  met  in  8  hours.  How  far  from  A's 
starting  point  did  they  meet? 

24.  A  man  found  that  he  could  row  2  miles  downstream  in  15 
minutes,  but  that  it  took  him  an  hour  to  row  back.  What  was  the 
rate  of  the  current?     Use  two  unknowns. 

25.  A  freight  train  leaves  a  certain  station  4  hours  before  a  pas- 
senger train.  If  they  travel  in  the  same  direction,  the  passenger 
train  will  overtake  the  freight  in  33  hours.  If  they  travel  in 
opposite  directions,  they  will  be  308  miles  apart  3  hours  after  the 
passenger  train  starts.     Find  the  rate  of  each.     Use  two  unknowns. 

26.  Two  autoijiobiles  are  225  miles  apart.  If  they  travel  toward 
each  other,  they  will  meet  in  5  hours.  If  they  travel  in  the  same 
direction,  the  faster  will  overtake  the  slower  in  45  hours.  Find 
the  rate  of  each. 

27.  The  length  of  a  rectangle  is  4  feet  more  th.an  twice  its  width. 
If  the  area  is  30  square  feet,  find  the  dimensions. 

28.  The  altitude  of  a  triangle  is  3  feet  less  than  twice  its  base. 
If  the  area  is  85  square  feet,  find  the  number  of  feet  in  base  and 
altitude.  ■ 


204 


BEGINNERS'  ALGEBRA 


29.  One  base  of  a  trapezoid  is  2  feet  more  than  the  other.  The 
shorter  base  is  twice  the  altitude.  If  the  area  is  21  square  feet, 
find  the  dimensions. 

30.  From  each  of  the  4  corners  of  a  square  piece  of  tin  a  square 
3  inches  on  a  side  is  cut  out.  If  the  total  area  remaining  is  540 
square  inches,  find  a  side  of  the  original  square. 


31.  A  piece  of  tin  is  20  inches  on  a  side.  How 
large  a  square  must  be  cut  from  each  corner  that  the 
area  remaining  shall  be  364  square  inches?  (Fig.  46.) 

32.  A  square  2  inches  on  a  side  is  cut  from  each 
of  the  4  comers  of  a  certain  square  piece  of  tin.  Fig.  46 
The  sides  are  then  turned  up  to  form  an  open  box.    Find  one 
side  of  the  original  square  if  the  volume  of  the  box  is  242  cubic 
inches. 

33.  A  certain  garden  is  50  by  40  feet  (Fig.  47).  It  is  divided 
into  4  parts  by  2  walks  running  through  the  center  parallel  to  the 
sides.  If  the  total  area  of  the  walks  is  261  square  feet,  find  the 
width  of  the  walk. 


12 


iW 


Fig.  47 


Fig.  48 


Fig.  49 


34.  A  little  park  is  100  by  75  feet  (Fig.  48) .  A  walk  is  around  the 
outside.  Find  the  width  of  the  walk  if  its  total  area  is  1464  square 
feet. 

35.  A  piece  of  tin  is  15  by  25  inches.  Squares  are  cut  out  of  the 
corners  so  that  the  area  remaining  is  275  square  inches.  Find  the 
size  of  the  squares  cut  out. 

36.  A  piece  of  iron  of  uniform  width  is  of  the  shape  and  dimen- 
sions shown  in  Fig.  49.  Find  the  width  if  the  total  area  is  56  square 
inches. 


37.  A  rectangle  is  3  feet  less  than  twice  the  width. 
is  170  square  feet,  find  the  dimensions. 


If  the  area 


FUNDAMENTAL  OPERATIONS  205 

To  find  interest,  multiply  the  principal  by  the  rate  and  the  result 
by  the  time  expressed  m  years.  This  rule  is  stated  as  a  formiila 
thus:  I  =  PRT. 

Use  the  preceding  formula  for  solving  the  following  problems: 

38.  Find  the  interest  on  $520  for  3  years  and  4  months  at  4  per 
cent. 

39.  How  much  money  must  you  have  in  the  bank  at  4  per  cent 
interest  to  receive  $100  each  year  as  interest. 

40.  What  principal  put  at  interest  at  5  per  cent  for  three  years 
will  yield  $1200  interest? 

41.  What  is  the  rate  of  interest  when  a  bank  pays  you  $10  interest 
every  six  months  on  a  deposit  of  $500? 

42.  How  long  will  it  take  $2500  to  yield  $450  if  put  at  interest  at 
6  per  cent? 

The  amount  is  the  principal  plus  the  interest.  The  formula  for 
the  amount  is,  then, 

A  =  P-\-I 
=P+PRT 
=  P{l-{-RT) 
Use  the  above  formula  for  the  following  problems: 

43.  How  long  will  it  take  $1000  to  amount  to  $1300  if  put  at 
interest  at  6  per  cent? 

44.  Solve  the  equation  /  =  Pi2r  for  P,  R,  and  T, 

45.  What  sum  of  money  will  amount  to  $1800  in  3  years  at 
4  per  cent? 

46.  At  what  rate  will  $200  amount  to  $235  in  5  years? 

47.  In  what  time  will  $450  amount  to  $540  at  4  per  cent? 

48.  A  man  has  $1200;  part  of  it  is  at  interest  at  4  per  cent  and 
the  remainder  at  6  per  cent.  The  sum  of  the  incomes  is  $56. 
Find  each  principal. 

49.  A  man  has  $1500;  part  of  it  is  at  interest  at  5  per  cent  and 
the  remainder  at  6  per  cent.  The  interest  on  the  part  at  6  per  cent 
is  $2  more  than  the  interest  on  the  part  at  5  per  cent.  Find  each 
principal. 


206  BEGINNERS'  ALGEBRA 

50.  A  man  has  $7700  at  interest,  part  at  5  per  cent  and  part  at 
6  per  cent.  How  much  has  he  at  interest  at  each  rate,  if  the  interest 
from  one  portion  is  equal  to  the  interest  from  the  other  portion? 

51.  Solve  the  equation  A=P-\-PRT  for  P;  solve  for  R;  solve 
for  T. 

An  angle  is  said  to  be  the  complement  of  another  angle  if  their 
sum  is  90°.  What  is  the  complement  of  an  angle  of  30°,  40°,  70°, 
55°?    Express  in  algebraic  language  the  complement  of  an  angle  x. 

An  angle  is  said  to  be  the  supplement  of  another  angle  if  the 
sum  is  180°.  What  is  the  supplement  of  the  angle  30°,  80°,  110°, 
135°?    Express  in  algebraic  language  the  supplement  of  the  angle  x. 

52.  The  supplement  of  a  certain  angle  is  26°  more  than  3  times 
its  complement.     Find  the  angle. 

53.  Three  times  the  complement  of  a  certain  angle  added  to 
twice  its  supplement  is  360°.     Find  the  angle. 

54.  Find  an  angle  such  that  its  supplement  is  3  times  its  com- 
plement. 

55.  Find  an  angle  such  that  its  supplement  shall  be  twice  its 
complement. 

56.  Find  an  angle  such  that  its  supplement  shall  be  n  times  its 
complement. 

[^ ^__ ^ 

^   ^        ^  6  ^ 

Fig.  50 

Fig.  50  represents  a  teeter  board.  It  is  called  in  physics  a 
lever.  There  are  other  kinds  of  levers.  F,  the  point  of  support, 
is  called  the  fulcrum.  The  distances  a  and  b  from  A  and  B  to 
F  are  called  the  lever  arms.  W  and  P  are  two  weights  placed 
at  A  and  B.     The  teeter  will  balance  if 

a-W  =  b-P 

U  a  =  5  feet  and  b  =  5  feet,  what  can  be  said  about  W  and  P? 

If  a  =  S  feet,  b  =  4:  feet,  and  W  =  50,P  =? 


FUNDAMENTAL  OPERATIONS  207 

If  a  is  greater  than  /;,  which  is  the  heavier,  W  or  P,  when  the 
teeter  just  balances?  If  F  is  at  the  middle  of  the  board,  what 
will  be  the  relation  of  the  two  weights?  Assign  values  to  W  or  P 
and  arrange  on  board  so  that  they  will  just  balance. 

57.  If  a  =  5  feet,  b  =  10  feet,  and  P  =  30  potinds,  what  is  the 
weight  of  W? 

58.  If  W=4:0  pounds,  P  =  30  pounds,  and  a  =  Q  feet,  how  far  is 
b  from  the  fulcrum? 

59.  Two  children,  W  and  F,  just  balance  each  other  on  a  16-foot 
teeter  board.  W  weighs  45  pounds,  F  weighs  75.  Where  is  the 
support  placed? 

60.  A  16-foot  teeter  is  balanced  at  the  center.  A  boy  weighing 
45  pounds  is  at  one  end.  How  far  from  the  support  must  a  boy 
weighing  60  pounds  sit  in  order  just  to  balance? 

61.  A  boy  weighing  45  pounds  sits  at  the  end  of  the  teeter  of 
Exercise  60.  Where  shall  a  boy  of  30  poimds  sit  in  order  to  balance 
the  first  boy? 

62.  A  20-foot  teeter  is  supported  at  a  point  4  feet  from  center. 
A  42-pound  boy  sits  at  the  end  of  the  longer  arm  and  just  balances 
a  boy  at  the  other  end.    How  much  does  the  second  boy  weigh? 


CHAPTER  IX 

Fractions 

151.  Definitions.  The  result  of  an  inexact  division  always 
contains  a  fraction.  The  arithmetical  fraction  f  is  the 
result  of  dividing  2  by  3.  The  fraction  -J  or  its  equal,  the 
mixed  number  2 J,  is  the  result  of  dividing  7  by  3. 

Any  division  may  be  so  expressed  whether  the  division 
be  exact  or  inexact. 

^  is  in  fractional  form,  though  the  result  of  the  division 

is  a  whole  number  or  integer. 

x  —  l 
In  the  same  way is  an  algebraic  fraction  expressing 

the  division  of  r!i;  —  1  by  :5^. 

This  suggests  that  the  most  satisfactory  way  of  defining 
a  fraction  is  to  say  that  it  is  a  certain  way  of  expressing 

division;   namely,  t-,  which  means  that  the  upper  number 

(numerator)  is  to  be  divided  by  the  lower  number  (denomi- 
nator) . 

152.  Historical.  We  do  not  know  when  men  first  used 
fractions,  but  it  must  have  been  long  before  the  time  of  the 
Babylonians,  for  we  find  fractions  on  clay  tablets  that  have 
been  dug  up  in  Babylonian  cities  buried  for  thousands  of 
years.  The  fractions  used  by  the  Babylonians  all  had  the 
same  denominator,  60.  The  early  Egyptians  used  unit 
fractions;  the  numerators  were  1,  the  denominators  differ- 
ent, though  they  did  have  a  special  symbol  for  f .  The 
Greeks  used  fractions  of  a  more  general  kind  with  any 
numerator  and  any  denominator.  None  of  these  peoples 
wrote  fractions  as  we  do.     (See  Fig.  51.) 

The  Babylonians  wrote  only  the  numerator,  the  denomi- 
nator being  understood,  and  the  fraction  being  indicated 

208 


FRACTIONS  209 

by  placing  the  numerator  a  little  to  the  right  of  the  ordinary 
position  of  the  number  in  the  line.     The  Greeks  placed  one 

BABYLONIAN  <<<    ^^^^   |§  '  ^ 

EGYPTIAN     CZZ  for  i  ,   c^  for  1,   CIJ>  for  i 

GREEK    [S' Ka"  for  ^ 

Fig.  51 

stroke  '  after  the  numerator,  and  two  strokes  "  after  the 
denominator.  The  Hindus  as  early  as  300  a.d.  wrote  the 
numerator  over  the  denominator,  but  with  no  line  between : 

8      ^"^      8 
The  line  between  was  introduced  later  by  the  Arabs  or 
possibly  by  the  Hindus.     It  was  certainly  used  before  the 
time  of  Columbus,  as  we  find  in  a  manuscript  on  algebra 
written  at  that  time  the  expression 

100         .         100  ' 

for      — 


1  ding 

153.  Laws  governing  the  use  of  fractions.  In  arithmetic 
we  combine  numbers  whose  exact  values  are  known;  in 
algebra  we  combine  numbers  without  knowing  their  special 
values.  The  rules  that  hold  in  arithmetic  must  also  hold 
in  algebra.  Algebraic  fractions  are  handled  according  to 
the  same  rules  of  operation  as  are  arithmetical  fractions. 
The  fundamental  ideas  are  the  same,  but  the  forms  of  the 
fractions  are  often  much  more  complicated. 

154.  The  fundamental  principle.  Fractions  may  be  put 
into  various  forms  without  having  their  values  altered.  A 
certain  fundamental  principle  must  be  kept  in  mind  in 
making  such  changes;  namely,  if  both  terms  of  a  fractionate 
divided  (or  multiplied)  by  the  same  number,  the  value  of  the 
fraction  is  unchanged. 


210  BEGINNERS'  ALGEBRA 


Illustration: 

3  3  •  5     15 

4  4-5     20 

2a  2ac 
Sb~3bc 

12     12^4    3 

28~28^4~7 

Sax  da 
bhx     56 

This  principle  may  be  put  into  symbols: 

an  _a 

hn    b 

The  second  fraction  is  the  first  fraction  with  its  terms 

divided  by  n;  the  first  fraction  is  the  second  with  its  terms 

multiplied  by  n.     Verify  this  relation  by  using  numbers  for 

a,  6,  and  n. 

155.  Reduction  of  a  fraction  to  its  lowest  terms.     How 

is  a  fraction  reduced  to  its  lowest  terms  in  arithmetic? 
(See  Art.  1.)  In  algebra  a  fraction  is  reduced  to  its  lowest 
terms  in  exactly  the  same  way  as  in  arithmetic.  Divide 
both  terms  by  the  same  number.  The  number  used  as  a 
divisor  must  be  a  factor  of  the  numerator  and  of  the  denomi- 
nator. 

49      JT-  7      7 


161    r-  23     23 

(1) 

ab    fib  _b 
'  ac    ^c    c 

(2) 

ab-^ac    0i{b-\-c)  _b-^c 
ad           /Kd           d 

(3) 


a{x-\-l)     a(*-p1jr    a 

EXERCISES 


Reduce  to  lowest  terms: 

1    ??                               2   2'2'7'2  3    186 

77                                •    2-3-2  *  306 

,    Sax                             -    Sax*  ^    5ax^ 

4.   „ —                              5.  ~= —  6.  --r- 

oay                                 I5ax  lobx 


FRACTIONS  211 

x^—xy  *  a^-{-ab 

11    ^Z^  12    ?§? 

•  a6-62  •  256 

j^       5a;c-5a  ^^    2'^*  3-  11 

15a2:r+20a  *      2^-7 

17     ^'+2^  IS    ^''-^r+l 

20  ^^^^y-^^y  21  •^''+^^+^ 

23.  ^±±  24.  ^^-'■^+'* 

*  10^)2 -15a2  •  3/-6 

156.  Multiplication  of  fractions.     How  is  the  product  of 

two  fractions  found  in  arithmetic?  (See  Art.  3.)  The 
product  of  two  fractions  in  algebra  is  found  in  exactly  the 
same  way: 


7. 

2^-9 
2^  •  3- 

10. 

Sx^ 

5x^-2x 

13. 

2aP 

16. 

19. 

189 
4-81 

22. 

x^-2x-{-l 

l-x^ 

25. 

Rr^-R^ 

ttR^  -  irrR 

In  arithmetic, 

In  algebra,  (1) 

(2) 


i   5^15 

7*4     28 
a  ^  X  _ax 
b    y     by 
Sx       2x         6x^ 


1    x-\-l      x^-l 


/QN  a         ax 

(4)    -^.(.+  1)  =  3£^ 


EXERCISES 


Multiply: 

l.|by5  2.^by8  3.  ^  by  5 

A     ^  u  e     2+JC  ,       -  ^     X—]    , 

4.  J  by  Of  5.^-^  by  5  6.  ^^^  by  or 


12  BEGINNERS'  ALGEBRA 

^-  ^3  ^y  ^+1  ^'  ^+2  by  ^             9.  y-3  by  ^ 

10.  Z. 2  11    22.2                               7^.5 

5    3  7     3                               3      2 

iq    2    5     4  ^.    2a    46  ^-    3a     5a 

''•3    7'n  ^^-  F*  5"  ^^'  2ft'7^ 

16    ?!^.Z!^  17       ^         3:y3  ^g      2:*;      2^+1 


3/       5/3                         a:+2     jc-2  *  x+1      x-2 

19      ^^    .  _^ .  £        20    i^^  •  ^+^  21    ^'+^       ^ 

*  x-j-1     x-l     X            '  x-2    x+1  '    x-S    x+2 

22.  ^^.2«3                23.-^ ^^±i-  24.^^.-2^ 

a-\-n                               x-l     x'^-i-x-\-i  x-i-l     x-l 

It  is  usually  desirable  to  reduce  the  result  of  a  multipli- 
cation of  a  fraction  to  its  lowest  terms : 

x-\       2x          J,x^^2x  2x 


x  +  1    x-l     (.v+D^^-'tT    ^+1 

25/ ^'-y'.     ^  26.  -^-^^^^ 

a        x  —  y  '  x-\-y         9 

27    ^izZ^  •  ^'+y'  28      ^'~^^    •  x2±2x-S 

'  x^-\-y'      x-y  '  x^-i-x-2    x^j-x-Q 

29    -^  •     ^'-^  oo      ^/ig       w^-/g^ 

The  dividing  out  of  the  factors  common  to  numerator 
and  denominator  may  be  done  before  the  multiplication  of 
the  two  fractions  if  desired.     This  is  often  the  better  plan: 

2.V       _2^ 

A  +  l 

„„  552  21n^s 
^^'  7^ '  l5^ 
q.    a+6       3 

39.    ^''  ^^-1 


X  +  l     -ifr^ 

31. 

2x2* 
4y'9y 

0^    3»  .  lOw/ 
^^-  5/        6 

34. 

a*    be 
b   'a 

or;    63     45 

^^•25'l4 

37. 

2x^        15a 
5           X 

38.  ^.^.| 
y    X    2 

x-l  2x 


FRACTIONS 


213 


40. 
42. 
44. 
46. 


irr    4r 

2x(x-2)     ^l(;c-^-:■<) 
x-\-.i        2(x-2) 

/3         K'-lrR+r^ 
3jc(.t-5)  :c2-25 


25 -r«       :v:4-f-2r''- 35:^2 
48.  ^'±i^    /,2 -13^+42 


50 


P  -1p  P^+2p 

e     f         ef 


e'  ■  ef    e2_^/2 


•  5{a-b)  '  ba+b^' 


43. 
45. 
47. 
49. 
51. 


^2+2^+3  ^  a:'^-2y-3 
x-S  x-2 

xr*     2r       a 
£^_5^3_|-6x2  ^  ^2-2:^- 15 

^2-9  :c3-4x 

iV3_SA^+15     m-15N±5Q 


A^2_i2iv+35    iV2-17A^+72 


a;2-2jc-3     2.T-2 


(^-3) 


157.  Division  of  fractions.  How  is  one  fraction  divided 
by  another  in  arithmetic?  (See  Art.  5.)  In  algebra  the 
process  is  just  the  same. 

b  '  y     b     X     bx 


x^-1     x+l 


X 

x^-1 


'  x*-±i 


x-l 


EXERCISES 


Perform  the  indicated  operations: 


^'  8    4 
3   ^^^ 

2a^Sc 
^'  3c  '  4a 
-    3a«_^6a^ 
'  •    56  '  106x' 

x''+7x+\2     :r+4 


9. 


11. 


:c2-4jc-4-4    *  x-2 
2(a-l)_4(l-a) 
3(«+l)  •5(:«-l) 


2. 


63^27 
32  "16 


'  x  '  x^ 


6. 


10. 


12. 


2ic 


x-l  '  x^-1 

a-^b  ab-\-a^ 
a^-ab'^  ah-¥ 
ic2+3x      a:+3 


2a        Sb 


a  —  b  '  a-\-b 


214  BEGINNERS'   ALGEBRA 


13.  -^-^ia-{-b)  14.  (a+6)H— ^ 

a—o  a—b 

15.  (:»;2-9a:+20)-^  16.  ^^^!!-^:^ 

x-\-b  16     20     12 

11^22     20  5y2^2h^^35o: 

25  '  35     13  7a'  "  4ax  *  ta^x 

/?^-n-20  .    n+1    .?^^+2^-8  7:y  .  21^^ 

W2-25      '  «2_25  ■  ^^w-2  ^^^-  3a  '    26 

^-ll/+30^/-5       /^-9  x^-1  .      a:-1 

"       /2_6/_|_9    •  ^2_3/^^2_36  ^^-     3;^;     •        2a 

158.  Addition  and  subtraction  of  fractions.     Just  as  in 
arithmetic  two  cases  must  be  considered: 

(a)  Fractions  with  the  same  denominator 
(6)  Fractions  with  different  denominators 
Exactly  the  same  methods  are  used  in  algebra  as  in 
arithmetic. 

159.  Fractions  having  the  same  denominators. 

4     5     2 

In  arithmetic ,  q + q  ~  o  ~  ^ 

O       Ct       o 

T-      ,     ,                        a  ,  h      c     a-\-h  —  c 
In  algebra,  — | 


State  as  a  rule. 


n     n     n 


EXERCISES 


1    9+2     1_5  1     2^5                    n     2jn  ^_n 

^-  7^7^7    7  "^^  3^3    3               '^'3^3  ^3 

,    2^  ,  5^    3ai;  _    7a  ,  a     4a           ^    I     a  ,  2a 

4   — 5.  — I (5, 1 — 

a      a      a  n     n     n                 x     x     x 

^    '3a-b     b  ^    3-b.b     2         ^^    2R-'Sr  .  2r-R 

in    3/>-^,   />-g  11    2a    2a -6 -f            ^^_^!±3 

A"-  2p-q^2p-q  ^^'    9             9            '*'•  :x;2+l     x^-\-l 

x^-3x-^2     2a;»+5a;-7  40:^-9  jt-=^-5    5-2;>:^ 

13.       ^3+1     +      ^^s+i  !*•    x^^l  x^+l^x^-{-l   ■ 


FRACTIONS  215 

160.  Fractions  with  different  denominators.  If  the  frac- 
tions to  be  added  have  different  denominators,  they  must 
be  reduced  to  equivalent  fractions  having  the  same  denomi- 
nator before  they  can  be  added.  In  many  instances  the 
required  common  denominator  can  be  found  by  inspection. 
(See  Art.  6.) 

T        :u      .-  2,1     4,3     7 

In  arithmetic,  ^'^'^Tfi^fi 

In  algebra.        (1)    -+-  =  -+_  =  ^_ 

b      n     bn     bn  bn 

Evidently  in  this  case  the  required  denominator  is  the 
product  of  the  denominators. 

bn  is  a  multiple  of  6,  why? 
bn  is  a  multiple  of  w,  why? 
bn  is  called  a  common  multiple  of  b  and  n. 

Any  number  that  can  be  divided  exactly  by  a  given 
nimiber  is  called  a  multiple  of  that  number.  The  common 
denominator  required  is  simply  a  common  multiple  of  the 
denominators.  Any  common  multiple  of  the  denominators 
of  the  fractions  can  be  used,  such  as  the  product  of  all  the 

denominators.     In  the  illustration  given,  the  fraction  7  is 

reduced  to  an  equivalent  fraction  whose  denominator  is  bn 
by  the  multiplication  of  both  terms  of  the  fraction  by  n. 


Thus, 

a     an 
b~b^ 

Illustration: 

X           1 

x-l     x-\-l 

x{x-\-l)     (x-l)]     x{x+l)-{x--l) 

X^-l           X''~l                    x^-1 

x^+x-x-\-l     x^-\-l 

x^-1  x^-1 


216  BEGINNERS'  ALGEBRA 


EXERCISES 

1    ?+  2    i+? 

-1-1  8.     3  2 


3C4-3  '  a;-2  a; -5    ic+3 


7    2    1 

3. 

4+5    3 

3a    26j_l 

♦>. 

ac      y   '  :*: 

9. 

a+6      2ah 

2       a+& 

12. 

2-+ii. 

a      J       c 

15. 

+     4- 
6c^ca^a6 

7     2     1 

18. 

9'^5~3 

21. 

X               X 

x-l     x'^-\-\ 

a—b    a-\-b  '  m-\-n    m—n 

a  —  1  a     0 

K K^  7a-b-2    2a-b+5 

^^-   l-e     1+e  •       a+4    "^     a -3 

19    -^— i-  20    '^±^—^ 

161.  Lowest  common  denominator.  Although  the  prod- 
uct of  the  denominators  of  all  the  fractions  to  be  added  will 
do  for  the  common  denominator,  it  often  saves  much  labor 
if  one  uses  the  smallest  denominator  that  will  accomplish 
the  purpose  sought.  An  example  will  best  illustrate  the 
point. 

First,  using  the  product  of  all  the  denominators  as  the 
common  denominator: 

_£ 1      ^         X{X+1) X^-1 

x^-1     x+1     {x^-l){x-\'l)      (:x:2_i)(^_|.i) 
x^+x-x^-\-l  ^         a; 4-1        ^     1 

(:^2_l)(;k;+l)       (^2_l)(^_|_l)       ^2_i 

Second,  if  it  is  noticed  that  x^—1  is  a  multiple  of  ^+1, 
the  work  would  proceed  thus: 

X  ^    _     ^         x  —  l_x—x-\-l_     1 

i^^     x-{-l~x^-l     x^-l~  x^-1    ~x^-l 

In  this  case  ic^  —  1  is  called  the  lowest  common  denominator 
of  the  fractions.  It  is  the  lowest  common  multiple  of  the 
denominators. 


FRACTIONS  217 

162.  The  lowest  common  multiple.  The  lowest  common 
multiple  (L.C.M.)  of  two  or  more  expressions  is  the  product 
of  all  the  different  prime  factors  of  the  expression.  Each 
factor  is  used  the  greatest  niunber  of  times  that  it  appears 
in  any  one  .expression. 

Find  L.C.M.  of  15,  12,  20. 


15  =  3-5 

12  =  22.3 

20  =  22-5 

L.C.M. 

is 

3- 5- 22  =  60 

Find  L.C.M. 

of 

x^-l,x^-{-2x+l,  and  a; 

a;2-1  =  (^-1)(%+1) 

x^+2x+l  =  (x-\-iy 

:v-l=x-l 

L.C.M. 

is 

(^+1)2(^-1) 

The  lowest  common  multiple  of  several  expressions  can 
be  found  by  inspection  of  the  factored  forms  of  the  expres- 
sions. The  factors  should  be  picked  out  according  to  the 
principle  given  in  the  definition. 

EXERCISES 

Find  the  L.C.M.: 

1.  36,  42,  12,  27  2.  5a^b^c^,  4abc^ 

3.  a^c,  hc\  cb^  4.  7a\  2ab,  Zb^ 

5.  2xy,  Zx^y,  4:xy^  6.  2^a^b'^c^,  42a^b^c^ 

7.  (x-y)2,  x-'-y^,  x^  8.  :x:-3,  «;-4,  :x;2_7^_|_12 
9.  l-2a,  14-2a,  4a2-l 

10.  x^-\-5x-\-6,  x^+Sx-]-2,  x^+4:X-\-Z 

11.  n^-b^,  n-^b,  in'-¥y  12.  6^2+54,  3A-9,  3^2-27 

13.  12x^-{-Sx-A2,  12x2+30a;-|-12,  32x2-40:«-28 

14.  ab-b\  ab-a\  {a-b),  a 

15.  x^-1,  2:c-2,  l-x,  x-1,  x 

16.  3(«-2),  w2-4w+4,  w2-2» 
15 


218  BEGINNERS'  ALGEBRA 

163.  Addition  of  fractions.  Add  the  following  fractions.  Be 
careful  to  remember  that  it  is  desirable  to  reduce  the  resulting 
fractions  to  lowest  terms. 


1. 

5+ A                         2 
4^12                          2- 

A_i.                 o  Z+A 

21     14                         "*•  8^12 

4. 

5-M                     5 
3    4^12                    ^• 

2    3                                  2     5 

7. 

n^2n 

8    ^+^ 

9. 

2               3a; 

x-\     (^-1)(:^+1) 

10       ^^        1       ^* 

11. 

3x         2x 
x2-l     x^-\ 

x-l                 x+3 

^-  (*+2)(x+l)     (:t+l)(*-3X 

13. 

x^l             x+3 
x2+5x+6    x2+3:r+2 

x-\-a       a;+3o 

15. 

3      2       3 

»"'"5w2     2n 

,„    4       3           2 
^''-  *    ^+1    {x^\Y 

17. 

2              5             1 

(x--3)3     {x-Zy^  x-Z 

,„       3*           7            2 
^*-  4*2-9' 2j;+3     2^-3 

19. 

2a+l     3a -1     a(a+3) 
a+1      a-\   '    a2-l 

2(:>:+3)     3x-6        4(*-l) 
"•  a:2-2j:       x^-Zx    x-'-hx-\-^ 

21. 

x-4       :*:-8 
x^-x     x^^r'^x 

22.  ^   .'^+'r 

a—x    a^—x^    a-tx 

23. 

2ax      x-\-a 
x^-a^    x-a 

^+3,     ^_3,     2x2_63,2 
'^^'  x-y    x-\-y^    x'-y^ 

25. 

1        1              1 

26   ?+     ^            ^ 

a     a-\-b     a^b—ab^ 

27. 

l+n-    ' 
1-n 

29. 

i+^+^'+^l^ 

•^•.+,+^    ^ 

31. 

l+x 

32.    1+X+X'  +  X'+:^^ 

33. 

.^+3-^^-2. 

„,    »(»-!)  ,«(«-l)(»-2) 
34-        2      "•            6 

FRACTIONS  219 


n(n-l),n(n-\-l)(n-l) 
6b.        2       "^  6 

n(n-l)(n-  2)     ni7t-l){n-2)in-S) 
^^^-  2-3  24 

37.  ..-i-.-.-.^—.  38.       ^^^         ^     '-' 


T^-2a^T-\-a'     a^  h{a-h)     c(2c-a-b) 

^^-  4(z2  4  ■      "•         2ib-c)         2{b-c) 

Find  a  short  way  of  adding : 

x  —  1     x-\-X     x-2     x-\-2 

43   -i 5_+_3 L_ 

^^'  x-S     x-l^x-\-l     ^+3 

45,  _fi_.  _i ^ 

a  — 6     aH-6     a  —  b  ^^'  a  —  b  '  &— a 

164.  Signs  of  fractions.  Exercise  46  of  the  last  article 
can  be  worked  in  a  simpler  way  if  one  takes  advantage  of 
a  certain  peculiarity  in  it. 

a     ^    6 


42      ^          ^    +    '    - 
*''■  x+1     x-1^  x+2 

1 
x-2 

1-n     1+w     l+M'^ 

^l+»* 

46.  -^+7— : 

a-b  '  6 


You  will  notice  that  the  denominator  of  the  second 
fraction  is  the  negative  of  the  denominator  of  the  first 
fraction.  (See  Art.  66.)  We  can  make  the  two  denominators 
the  same  by  multiplying  both  terms  of  the  second  fraction 
by  -1. 

Thus,  -^4 


or 


a  —  b     —b-\-a 
a         —b    , 
^~^^a-b 


The  sum  is  thus  r  =  1 

a  —  b 


This  illustration  leads  to  the  consideration  of  the  changes 
in  sign  that  can  be  made  in  the  terms  of  a  fraction. 


220  BEGINNERS'  ALGEBRA 


We  know  that  r-  =  r 

on    b 

is  true  for  all  values  of  n  except  zero. 

It  is  true,  then,  for      n=—l 

a  •  — 1_  —a 

a     -a    2     -2 

J        .  — aX  — 1       a 

and  again 


bX-l       -b 

That  is,  -7—  =  — r 

0       —0 

Hence  one  can  say  the  signs  of  both  terms  of  a  fraction 

can  be  changed  without  altering  the  value  of  the  fraction. 

Furthermore,  we  know  by  the  law  of  signs  for  division  that 

the  quotient  of  a  negative  number  and  a  positive  number  is 

a  negative  number: 

— a_     a 

T~~~b 

and  also  -r-  =  — r 

—0  o 

and  we  may  then  write  the  very  interesting  identities 

a   _  —a_     a_      —a 

^^~~b'~~b~~"^ 

The  significance  of  these  identities  will  be  seen  more 

clearly  if  the   +   signs  are  introduced  where  they  might 

properly  be  placed : 

'^~^b~  "^T^~  ~+&~  ~~^ 
Speaking  of  the  three  signs,  the  sign  of  the  numerator, 

the  sign  of  the  denominator,  the  sign  of  the  fraction,  wc 

may  say  that  if  any  two  of  them  are  changed  the  value  of 

the  fraction  is  unchanged. 

What  would  be  true  if  only  one  of  these  three  signs  were 

changed? 


FRACTIONS  221 

These  changes  of  sign  in  a  fraction  are  very  convenient 
in  altering  the  forms  of  fractions  so  that  they  'may  be  more 

readily  combined. 

1-x 
x^-l 
can  be  put  in  the  more  convenient  form 

x-1  1 


The  fractions 


x^-1        x+1 
1  4 


x—l     1—x 
can  be  more  readily  combined  if  we  change  the  form  to 

The  usefulness  of  these  changes  in  sign,  of  course,  depends 
upon  your  ability  to  recognize  instances  where  such  a 
change  can  be  applied. 

EXERCISES 

_^__2x_  2  1 


:»-2     2-jc  •  a;2-l     1-a; 

3    _2 1 l_  ^      a     jg-fl L- 

1—x^     1—x    x  —  l  '  a—b     b—a     a—b 
1          4          6  x-\-y,     X 

O.  ; — :; -7—, —  D.  "1 

a—1     1—a     l+fl  x—y    y—x 

7.  °      .+J-  8.     "     ■     "' 


a*— 3a— 4    4— a  a  —  b    ba—a^ 

165.  Complex  fractions.  When  the  division  of  fractions 
or  expressions  containing  fractions  is  written  in  the  frac- 
tional form,  the  whole  expression  is  called  a  complex  fraction. 
Several  of  the  examples  of  Art.  157  can  be  written  in  this 
form. 

be  written       b^ 


W^  J     \b'     V     inoi 
This  latter  form  is  a  complex  fraction 


one  form  ^_i 


222  BEGINNERS'  ALGEBRA 

Complex  fractions  can  often  be  reduced  to  the  form  of 
simple  fractions  rhost  readily  if  we  make  use  of  the  funda- 
mental principle  of  the  fraction  that  both  terms  of  a  frac- 
tion may  be  multiplied  by  the  same  number  without  any 
change  in  the  value  of  the  fraction. 

In  the  case  given  above  multiply  both  terms  by  6^: 


(1) 

So  also, 
(2) 


-+1 


«-') 


1-1   rif_,V  " 


2+-     a6(2+j) 


1_ 

a      "*^^\"'  '  a  J     2ab-{-b 


1     1         ,/l     A       a  —  ab 
^-1       ab( 


"G-0 


.   ,^,                2"^3      V2"^3r       3+2  5 

The   number   used  as  a  multiplier  is  the  L.C.D.  of  all 
fractions  that  appear  in  the  terms  of  the  complex  fraction. 

EXERCISES 

Reduce  to  simple  fractions: 

1+1                                 1_3  «±2+    « 

I          X                             2    ^  3       g       a— 6 

X                                      ' a  a-S 

aj)_                                 i_^zi_  1_1 

4.  L^                              5.  _   2_-  6.  2-^ 

y'a                                   1        2        ^  2"'"3 


FRACTIONS 


223 


32 
5     3 


9.  Evaluate 


8.  Evaluate 


a  —  b 
l-\-ab 


for  a  =  3,ft  =  ^ 


10. 


l-x 


fl 


for  F  =  231,  yl=2,  ^  =  3,  C  =  4. 


11. 


13. 


I -a 
1+a 


-1 


l-g 

l+d" 

1 


1  + 


y+1 
n—q 


12. 


14. 


1-a 

1+a' 


l+£ 

l-a 
1 


1 


i^+^ 


15.  Evaluate 


R^-{-2Rr+r^ 


x^+7 


16.  Evaluate 


3f2— 5jt;+6 


17. 


for  R  =  \,  r=\ 

1      .     1 


Evaluate 
for  6  = 


h—a     b—c 

2ac 


18.  Evaluate  -^-f     ^ 


x-\-6     ac+1     2a;     a; 


for  a;  =  4 


a-\-c 


19.  Evaluate  ^*  = 


21.  Evaluate 


20.  Evaluate  P  = 

forz;  =  3,     F  =  5 

2-;c 


1— «w 


22.  Find  value  of 


x-i-l 


for  n 


^    1-       417^- 


when  .k:  = 


2-a 
a-\-l 


23. 


Find  value  of 24.  Find  value  of  -^t_ 

x  —  o         X  V 


when  x  =  '- 


,             abc+ac         ac-\-a 
when  a;  =  —, ,  y  - 


b-c    ' 


b-c 


224 


BEGINNERS'  ALGEBRA 


166.  Graphs.     The  graphs  of  fractions  have  interesting 

12 


pecuUarities.     For  example,  consider  the  fraction 


12 
Make  a  table  of  the  values  of  —  from  %  = 

X 


12  to  :t  =  12. 


X 

12 

X 

X 

12 

X 

X 

12 

X 

X 

12 

X 

12 

1.0 

6 

2.0 

—1 

-12 

—  7 

-1.7 

11 

1.1 

5 

2.4 

—  2 

—   0 

-8 

-1.5 

10 

1.2 

4 

3.0 

—  3 

-   4 

-9 

-1.3 

9 

1.3 

3 

4.0 

-4 

-   3 

-10 

-1.2 

8 

1.5 

2 

6.0 

-5 

-2.4 

-11 

-1.1 

7 

1.7 

1 

12.0 

-6 

-2.0 

-12 

-1.0 

As  we  cannot  divide  by  0 

In  the  figure 
the  graph  is 
drawn  for  the 
points  on 
the  right  of  the 
vertical  axis. 
The  other  points 
should  be  joined 
in  a  similar  way. 

Starting  with 
ic  =  12,  what  is 
true  of  the  value 

of  —  as  X  gets 

X  ^ 

smaller  ?     What 
about  —  when  x 

X 

gets  very  small  ? 


12 


has  no  value  when  x  =  0. 


Fig.  52 


EXERCISES      • 

12 
1.  Plot  -r  in  a  similar  way. 

In  drawing  the  graphs  for  the  following  fractions,  draw  smooth 
curves  between  the  points  plotted.     Use  integral  values  for  x,  but 


FRACTIONS  225 

if  in  some  places  the  points  do  not  come  near  enough  together  use 
a  fractional  value  of  x  in  between.  For  the  most  part  let  1  inch 
on  the  axis  represent  1 . 

2.  ..+4  3.  -i5-  4.  J2^^ 

5.  ^±1  6.  J^,  7.  ^=1 

X  X^—4:  X 

It  is  important  to  notice  that  the  graphs  of  fractions  are 
very  different  from  the  graphs  of  integral  expressions.  We 
have  seen  that  the  graphs  of  expressions  of  the  first  degree 
are  all  straight  lines.  The  graphs  of  expressions  of  the 
second  degree  are  all  curves,  but  are  all  of  the  same  kind. 
The  graphs  of  fractions  include  a  great  variety  of  ctirves, 
some  of  which  are  shown  in  the  exercises  given  here. 

RATIO 

167.  Definitions.  The  quotient  of  two  numbers  or  of 
two  algebraic  expressions  is  often  called  the  ratio  of  the  two 
numbers  or  expressions.     (See  Art.  9.) 

The  quotient  7  is  called  the  ratio  of  a  to  6  and  is  some- 
times written  in  the  form  a:b,  though  generally  the  fractional 
notation  is  to  be  preferred.  The  numerator  a  is  often 
called  the  antecedent,  the  denominator  the  consequent. 
The  reason  for  these  names  is  evident  from  the  old  notation 
a:b,  the  one  that  goes  before  and  the  one  that  follows. 

As  a  ratio  is  simply  a  quotient,  it  has  all  the  properties 
of  a  quotient  and  is  to  be  treated  as  such.  (See  Art.  9.) 
The  term  ratio  is  used  so  frequently  in  mathematical  prob- 
lems that  you  should  become  thoroughly  acquainted  with 
its  meaning. 

EXERCISES 

1.  Express  the  ratio  of  4  to  6. 

2.  Express  the  ratio  of  3  inches  to  2  feet. 


226 


BEGINNERS'  ALGEBRA 


3.  What  is  the  ratio  of  the  length  to  the  width  of  a  rectangle  of 
dimensions  3  feet  by  7  feet? 

4.  Is  the  ratio  of  14  feet  to  2  feet,  7  feet? 

5.  Express  the  ratio  of  3  to  12  in  the  form  of  a  decimal? 

6.  Find  the  ratio  of  2  to  7  correct  to  3  decimal  places. 

7.  Measure  the  length  of  a  piece  of  paper  in  both  English  and 
metric  standards,  and  find  the  number  of  centimeters  in  1  inch. 
What  is  the  ratio  of  an  inch  to  a  centimeter? 

8.  What  is  the  ratio  of  a  mile  to  a  kilometer?  Of  a  kilometer  to 
a  mile?     A  kilometer  is  1000  meters.     A  meter  is  39.37  in. 

9.  In  1900  the  population  of  the  United  States  was  76,304,799, 
while  in  1890  it  was  62,622,250.  Find  the  ratio  of  the  population 
in  1900  to  that  in  1890  correct  to  .01. 

10.  The  density  of  population  is  the  ratio  of  the  population  to 
the  area;  that  is,  it  is  the  number  of  people  to  the  square  mile. 
Find  the  density  of  population  correct  to  one  decimal  place  for 
the  following  countries,  and  show  the  results  graphically  by  means 
of  a  series  of  straight  lines  of  the  proper  length: 


Country 

Area 

Population 

United  States 

Great  Britain .... 
France 

3,574,658 

121,633 

207,054 

1,802,629 

3,913,560 

91,972,266 
45,516,259 
39,602,258 

India  ... 

315,156,396 

China 

320,650,000 

11.  In  a  lot  of  7  dozen  eggs  the  ratio  of  good  to  bad  is  2  to  3. 
How  many  good  eggs  are  there  in  the  lot? 

12.  Find  two  numbers  in  the  ratio  of  7  to  8  whose  sum  is  90. 

13.  Which  is  the  greater  ratio,  -f  or  f  ? 

14.  What  is  the  ratio  of  -  to  -  ? 

0       a 

13.  What  is  the  ratio  of  -  to  -  ? 
X       y 

10.  Find  the  ratio  of  br'  to  ar^. 


17.  What  is  the  ratio  of to 


18.  Find  the  ratio  of  — — -  to 


FRACTIONS  227 

3a     ^ 


(x-ay 


«2-l      {n-iy 

19.  A  rectangle  is  3  by  5;  squares  are  constructed  on  one  end 
and  on  one  side.     What  is  the  ratio  of  the  areas  of  the  squares? 

ork    T^-   J  ^1         ^-       rn{n  —  l).     n(n  —  l){n—2) 

20.  Find  the  ratio  of  -^-^ — -  to  -^ ^^ -- 

21.  If  Sx—4:y  =  5x-^6y,  find  the  ratio  of  x  to  y. 

22.  In  what  ratio  should  nuts  at  13  cents  a  pound  and  27  cents 
a  pound  be  mixed  to  make  a  mixture  worth  20  cents  a  pound? 

23.  In  what  ratio  should  two  kinds  of  coffee  worth  20  cents  and 
30  cents  a  pound  respectively  be  mixed  to  make  a  blend  worth 
26  cents  a  pound?  « 

VARIATION 

168.  Definitions.  What  is  the  circumference  of  a  circle 
if  its  diameter  is  2  feet?  3  feet?  4  feet?  n  feet?  What 
is  the  ratio  of  the  circumference  to  the  diameter  in  each 
case?  If  the  diameter  is  doubled,  what  is  the  effect  on  the 
ratio  of  circumference  to  diameter?  If  the  diameter  is 
halved,  what  is  the  effect?  What  effect  will  any  change  in 
the  diameter  have  on  the  ratio  ?  If  the  diameter  is  changed, 
what'will  be  the  effect  on  the  circumference? 

Any  quantity  that  may  take  on  various  values  is  called 
a  variable  quantity.  The  diameter  of  a  circle,  the  speed  of 
an  automobile,  the  distance  a  train  travels,  the  height  of  a 
boy,  the  price  of  coal,  are  illustrations  of  variable  quantities. 

The  number  of  cents  in  a  dollar,  the  distance  between  two 
mile  posts  on  a  railroad,  the  number  of  days  in  January,  the 
number  9  or  6,  are  qua,ntities  which  do  not  change.  Such 
numbers  are  called  constants. 

When  one  number  changes  as  another  number  changes  so 
that  the  ratio  of  two  variable  quantities  is  constant,  that  is, 
is  always  the  same  number,  we  often  say  that  one  varies  as 
the  other,  or  that  the  one  is  proportional  to  the  other. 


228  BEGINNERS'  ALGEBRA 

We  write  the  statement  x  varies  as  y  in  the  form 
x_ 

y~ 

where  c  is  the  unknown  constant.     The  more  common  way 
of  writing  the  statement  is  without  the  use  of  fractions: 

x  =  cy 
which  indicates  another  way  of  wording  the  statement.  If 
one  quantity  varies  as  another,  it  is  some  constant  number 
times  the  other.  We  say  the  circumference  of  a  circle 
varies  as  its  diameter.  In  this  case  we  happen  to  know  the 
constant  and  write 

C=TrD 


PROBLEMS 

1.  The  price  paid  for  eggs  at  35  cents  a  dozen  varies  as  the  num- 
ber of  dozen  purchased.  What  is  the  constant  in  this  case?  State 
the  relation  in  symbols,  using  c  for  total  cost  and  d  for  nimiber  of 
dozen. 

2.  A  train  traveling  50  miles  an  hour  leaves  Chicago  for  New 
York.  Using  d  for  distance  from  Chicago,  and  /  for  number  of 
hours,  state  the  relation  between  d,  t,  and  50  in  terms  of  variation 
and  write  the  statement  in  symbols.    What  is  the  constant  ratio? 

3.  The  circumference  of  a  circle  varies  as  its  radius.  What  is 
the  constant  in  this  case? 

4.  The  diameter  of  a  circle  varies  as  the  circumference.  What 
is  the  constant  in  this  case? 

5.  The  amount  of  money  I  receive  for  12  dozen  eggs  will  vary 
as  the  price  per  dozen.    State  in  symbols. 

6.  The  area  of  a  circle  is  given  by  the  formula  A^irr^.  The 
area  varies  as  what? 

7.  The  volume  of  a  rectangular  box  varies  as  the  product  of  its 
three  dimensions.  State  in  algebraic  terms  and  state  what  value 
the  constant  has  in  this  case. 

8.  The  volume  of  a  cube  varies  as  what?    What  is  the  constant? 


CHAPTER  X 

Square  Roots  and  Quadratic  Equations 

169,  Square  root.    One  of  the  two  equal  factors  of  a 
niimber  is  called  the  square  root  of  the  number. 

3-3  =  9 

3  is  the  square  root  of  9, 
but  -3  •  -3  =  9 

Hence  —3  is  also  a  square  root  of  9. 

9  has  two  square  roots,  +3  and  —3,  which  are  numerically 
equal,  but  opposite  in  sign. 

We  use  the  symbol  V9  to  represent  the  positive  square 
root  and  —  V9  to  represent  the  negative  square  root: 

V9  stands  for +3 
-V9  stands  for  -3 
±  V9  stands  for  both  ±3 


EXERCISES 

I.  Vie 

2.   -V25 

3.  +V49 

4.  2V36 

5.  V^ 

6.   -V^2 

7.  ±yJ^ 

8.  +V4x« 

9.   -^9x' 

.0.  ±Vl6^' 

11.  Vl21 

12.   -VlOO 

.3.  ±\limf- 

14.  Vl44o2 

15.  Vl69 

The  symbol  V  ,  called  the  square  root  sign,  or  the  radical 
sign,  has  been  used  since  the  early  part  of  the  sixteenth 
century,  when  it  was  introduced  by  Riese  and  Rudolil. 
Before  that  time  a  number  of  different  symbols  were  used  to 
denote  a  square  root. 

229 


230  BEGINNERS'  ALGEBRA 

170.  Finding  the  square  root  of  a  number.  The  square 
root  of  a  number  may  be  found  in  several  ways.  Two 
methods  will  be  suggested  here: 

(a)  The  division  method  used  in  arithmetic  (see  Art.  220). 

(6)  The  use  of  tables  and  a  trial  method. 

The  following  table  of  squares  can  be  easily  calculated : 


No. 

Square 

No. 

Square 

No. 

Square 

No. 

Square 

1... 

...    1 

9... 

...81 

16.. 

....? 

24. 

? 

2... 

...  4 

10... 

. . . 100 

17.. 

,  .  .  .? 

25. 

..,..? 

3... 

...  9 

11... 

...121 

18.. 

....? 

26. 

? 

4... 

...16 

12... 

. . . 144 

19.. 

.  .  .  .  ? 

27. 

? 

5... 

...25 

13... 

. . . 169 

10.. 

? 

28. 

? 

6... 

...36 

14... 

. . . 196 

21.. 

....? 

29. 

.....? 

7... 

...49 

15... 

...225 

22.. 

....? 

30. 

? 

8... 

...64 

23.. 

....? 

Show  how  to  find  from  the  table  the  square  roots  of  81, 
169,  196,  484,  625,  729.     ' 

171.  Square  root  by  trial.  It  is  a  simple  matter  to  find 
the  square  root  of  any  number  that  appears  as  a  square  in 
the  table.  But  how  shall  we  find  the  square  root  of  a 
number  that  is  not  found  as  a  square  in  the  table? 

How  shall  we  find   yQ  ? 

Recalling  the  fact  that  a  square  root  of  a  number  is  one 
of  the  two  equal  factors  of  the  number,  we  are  simply  to 
find  a  number  which  multiplied  by  itself  will  equal  6.  A 
glance  at  the  table  will  show  that  since  6  is  between  4  and 
9  the  required  square  root  will  be  between  2  and  3,  for 
2 -2  =  4  and  3 -3  =  9 

We  may  take  for  trial  any  number  between  2  and  3,  but 
since  6  is  about  midway  between  4  and  9  it  might  be  well  to 
try  2.5. 

Now  2.5X2.5  =  6.25 

As  this  product  is  more  than  6  2.5  is  a  Httle  too  large; 
try  2.4. 

Now  2.4X2.4  =  5.76 


SQUARE  ROOTS  AND   QUADRATIC  EQUATIONS    231 

As  5.76  is  less  than  6,  2.4  is  a  little  too  small. 

The  factor  or  square  root  sought  must  then  lie  between 
2.4  and  2.5.  This  shows  that  its  first  two  digits  are  2.4. 
We  may  get  a  little  nearer  the  required  number  by  another 
similar  trial.  Try  2.45:  2.45X2.45  =  6.0025,  which 
shows  that  2.45  is  a  little  too  large.  Try  2.44: 
2.44X2.44  =  5.9536,  which  shows  that  2.44  is  too  small. 
The  root  then  lies  between  2.44  and  2.45.  The  first  three 
digits  are  certainly  2 .  44. 

This  process  of  trial  can  be  carried  out  as  far  as  one 
desires.  It  is  a  fact,  the  truth  of  which  will  be  shown  to 
you  in  a  later  course  in  algebra,  that  no  matter  how  long 
the  process  is  continued  an  end  can  never  be  reached  of 
finding  the  square  root  of  6.  The  string  of  digits  would  go 
on  forever.  The  exact  V6  can  never  be  found  in  the  form 
of  a  decimal  number.  The  number  found  at  any  stage  of 
the  calculation  is  called  an  approximate  value  of  the  root. 
For  our  purposes  it  is  not  necessarv  to  find  more  than  three 
digits  of  such  roots. 

If  the  process  be  applied  to  numbers  that  are  squares,  the 
work  would  come  to  an  end,  and  an  exact  square  root  would 
be  found.  It  will  be  seen  upon  reference  to  the  table  of 
squares  on  page  230  that  most  of  the  integers  between  1 
and  1000  are  not  squares.  Only  approximate  square  roots 
can  be  found  for  these  integers. 

Square  roots  of  numbers  that  cannot  be  expressed  exactly 
in  decimal  form  are  called  irrational  roots. 

EXERCISES 

1.  Find  the  square  roots  of  1,  2,  3,  10,  21. 

2.  Form  a  table  of  the  square  roots  of  the  integral  numbers  from 
1  to  10  and  preserve  for  future  use. 

172.  Number  of  square  roots.  Every  positive  number 
has  two  square  roots  equal  in  numerical  value,  one  positive 
and  the  other  negative.  What  are  the  square  roots  of  25, 
36,  49,  121,  5,  7? 


232  BEGINNERS'  ALGEBRA 

173.  Square  root  of  a  fraction.     The  square  root  of  a 
fraction  may  be  found  thus:  _ 

Since      ^  •  ^  =  q,  it  follows  that       \q'^q- 

2  is  the  square  root  of  4.     3  is  the  square  root  of  9. 
This  suggests  the  rule 


_        _       _    V^ 
n.1.  /5     V5      V5     2.23       ^,  , 

^^^^'    ^r^^  3=— =-^4+ 

fT  vr    2 


V4_V4  ^ 
3~V^~1 


=  1.1  + 


VS"     1-73 

Another  way  is  to  do  the  division  first  and  then  find  the 
root. 


4=^^= 


Thus,  -v/^=Vl. 5  =  1.2+ 

Other  methods  will  be  given  in  a  later  chapter.  But 
what  is  given  here  is  sufficient  for  any  cases  that  may  arise. 

EXERCISE 

„.    ,  ^,  ^      f  36   121    81     49     5     7    3  23  4   16 

Fmd  the  square  roots  of  ~,  — ,  — ,  — ,  -,  ~,  -,  -,  -,  ^, 

49   25  2  5  3  2 
3  '  6  '  3'  r  5'  5* 

174.  Quadratic  equations.  An  equation  of  the  second 
degree  in  one  unknown  is  called  a  quadratic  equation.  In 
previous  chapters  we  have  solved  equations  of  the  second 
degree  by  factoring.  But  there  are  equations  that  cannot 
be  factored  by  any  of  the  methods  you  have  studied  so  far. 
For  example,  you  cannot  at  this  time  solve  the  equation 

by  factoring.  It  can  be  done,  but  the  method  is  rather 
complicated.  It  is  better  to  use  a  simpler  method  if  one 
can  be  found. 


SQUARE  ROOTS  AND  QUADRATIC  EQUATIONS    233 

175.  The    square-root   method  of   solving   a   quadratic. 

Consider  the  equation 

or,  in  a  form  better  suited  for  our  purpose, 

a;2  =  9 

This  equation  asks  the  question:  What  is  the  number 
whose  square  is  9  ?  The  required  ntunber  is  the  square  root 
of  9,  or,  in  symbols, 

;c  =  ±V9 
x  =  3±,  that  is,    +3  and  —3 

We  have  taken  the  square  root  of  both  sides  of  the  equa- 
tion. 

It  is  unnecessary  to  use  the  double  sign  on  both  sides 
and  write  dLx  =  ±3 

for  this  would  mean  that 

(1)  +^=+3  (3)     -x=+3 

(2)  +x=-S  (4)     -x=-d 

in  which  the  third  is  the  same  as  the  second  and  the  fotuth 
is  the  same  as  the  first.     Why? 

The  work  given  above  discloses  another  operation  that 
can  be  appHed  to  an  equation;  namely,  the  square  root  of 
each  side  of  an  equation  may  be  taken  so  long  as  both 
roots  are  used. 

176.  Application  of  the  square-root  method.  Equations 
of  the  type 

a;2-49  =  0 

may  be  solved  equally  well  by  the  factoring  or  by  the  square- 
root  method.     The  square-root  method  leads  at  once  to  the 
solution  of  an  equation  like  x^—5  =  0,  which  is  not  factorable 
in  the  sense  in  which  you  have  understood  factoring: 
a;2-49  =  0  x^-5  =  0 

^2  =  49  x^  =  5 

x  =  ±^W  x  =  ±^'b 

x  =  ^^x=-l  x  =  2.23,  ::c=-2.23 

16 


234  BEGINNERS'  ALGEBRA 

EXERCISES 

Solve  and  check,  using  the  square-root  method:  ' 

1.  /  =  81  2.  :^2=i2l  3.  x^-SQ  =  0 

4.  4a;2=64  5.  2:^2-50  =  0  6.  144-/2  =  0 
7.  0  =  /2-100                   8.  x^  =  7  9.  y2_3  =  o 

10.  4x^  =  5  11.  9a2-49M)  12.  (:r+l)2  =  25 

13.  (:r-2)2=9  14.  (jc+5)2=l  15.  (w-l)2=16 

177.  Standard  form.  The  standard  form  of  the  quadratic 
equation  is 

ax^-\-bx-^c  =  0 

Write  equations  that  come  under  this  form  by  writing  in 
values  for  a,  b,  and  c.  Quadratic  equations  can  be  reduced 
to  this  form  by  collecting  the  terms  of  the  same  degree  and 
arranging  them  in  the  order  of  the  powers.  It  is  generally 
desirable  to  put  the  equation  to  be  solved  in  the  standard 
form.  In  some  cases,  however,  this  would  be  foolish.  Would 
it  be  at  all  desirable  to  put  the  following  into  the  standard 
form  in  order  to  solve  them?     Why? 

,      (a-sy  =  ^,     n(w+2)=0 

EXERCISES 

Put  the  following  equations  in  standard  form: 

1.  2x^-Sx-\-2  =  5x-S  2.  S-2x  =  5x-2+S 

3.  5{n-2)=3n{n-l)  4.  7x-2^Sx{x-2)=4x^-S 

5.  3:«2_5  =  2x(3-.t;)+2^  6.  7(«-2)2-3(»+l)  =  7«-f«« 
7.  (w+2)(n-3)  =  (w-l)(«+3)    8.  x{x-l)-4(x-S)  =  2x{x-2) 

178.  Square-root  method,  completing  the  square.  The 
same  method  may  be  applied  to  an  equation  in  which  the 
first-degree  term  is  present. 

For  example,  ^^  _  g^  _  10  =  0 

Arrange  the  equation  so  that  the  terms  containing  the 

unknown  shall  be  on  one  side  and  the  known  terms  on  the 

other: 

x'-Qx=^m 


SQUARE  ROOTS  AND  QUADRATIC  EQUATIONS    235 

Make  the  side  containing  the  unknown  a  trinomial  square 
by  adding  the  proper  term.     (Review  Art.  118.) 

rv:2-6xH-?  =  16+?  (1) 

/6\2 

In  this  case  add  9,  9  =  (^^  )  (2) 

x^-Qx-\-9  =  m-\-9  (3) 

=  25 

Show  the  square  in  another  form  * 

(x-3)2  =  25  (4) 

x-S  =  ±5  (5) 

:v  =  ±5+3  (6) 

:^  =  8and  -2  (7) 

It  is  necessary  to  add  the  two  numbers  on  the  right-hand 
side  before  taking  the  square  root. 

This  method  is  called  the  square-root  method  of  com- 
pleting the  square. 

If  the  coefficient  of  the  term  of  second  degree  is  some 
other  number  than  one,  the  terms  of  the  equation  should 
be  divided  by  that  coefficient.  This  makes  it  easier  to  see 
what  must  be  added  to  complete  the  square. 


3«2-2«  =  16 

(1) 

Divide  by  3, 

*      3*-3 

(2) 

A^^  (i-i)" 

■    2'     ,   1     16  ,  1 

*^-r+9=3-+9 

(3) 

Change  form, 

/        1 \2     49 

(*-  3)  -  ¥ 

(4) 

Take  square  root, 

1      J." 
^-3=^3 

(5) 

Solve  for  x, 

,7,1                   7.1 
*=+3+3        *=-3+3 

(6) 

=  1 

(7) 

236  BEGINNERS'  ALGEBRA 

Rule.  (1)  Reduce  equation  to  the  standard  form 
ax^-\-bx  =  c 
by  collecting  all  terms  containing  the  unknown  on  one  side 
and  the  known  terms  on  the  other.  And  if  necessary 
divide  each  side  by  the  coefficient  of  the  term  of  second 
degree. 

(2)  Complete  the  square  of  the  side  containing  the 
unknowns  by  adding  to  both  sides  the  square  of  half  the 
coefficient  of  the  term  of  first  degree. 

(3)  Take  the  square  root  of  both  sides. 

(4)  Solve  the  two  resulting  linear  equations  for  the 
unknown. 

EXERCISES 

Solve  by  completing  the  square: 

1.  x^-'2x  =  ^8  2.  /-6y-40=0 

3.  x^-\-8x=Q  4.  a;2+10:c-ll=0 

5.  /2+14/+24  =  0  6.  x^-3x  =  4: 

7    aM-a-6  =  0  8.  x^-^X'>r3  =  0 

9.  5+^2 -6x  =  0  10.  y^+l2y+27  =  0 

11.  x2-5x+6  =  0  12.  '2x2-7jt;=15 

13.  3y2=14-19y  14.  7x-^5  =  Qx' 

15.  ^x^-2x=G-2x^-{-3x  16.  6^2+6=  13jc 
17.  i5x-\-2)x=S{l-x^)       -  18.  5x2 -3:3c+10  =  9 -10x2 -lire 

179.  More  difficult  exercises.  Let  us  try  the  square- 
root  method  on  equations  that  cannot  be  factored. 

:^2-4:^+2  =  0 
Change  the  form,  x^—4x=  —2 

Complete  the  square,  ic^ — 4^ + 4  =  2 
Put  in  square  form ,  (:^ — 2)  ^  =  2 
Take  square  root,  ic-2=V2^  x-2--^ 

^-2==+1.41,    :t-2=-1.41 
Solve  for  rx;,  ^  =  3.41  ;c=.59 


SQUARE  ROOTS  AND  QUADRATIC   EQUATIONS    237 


Check  for  ^  =  3.41: 

3.4P-4(3.41)+2    0 
11.6281-13.64+2    0 
-.0119    0 
The  two  sides  do  not  come  out  the  same,  but  the  error 
is  very  small  and  is  due  to  the  fact  that  1.41  is  only  an 
approximate  value  for  V2:  consequently  3.41  is  merely  an 
approximate  value  for  the  number  sought.     Its  exact  value 
cannot  be  expressed  in  decimals. 

Recalling  Art.  15  on  the  use  of  approximate  numbers, 

we  might  have  done  the  checking  thus: 

3.4P-4(3.41)+2    0 

11.6-13.6+2    0 

0    0 

and  for  .59,  .592-4(.59)+2    0 

.35-2.36+2    0 

-.01    0 

Take  another  equation  for  illustration : 


Divide  both  sides  by  3, 
Complete  square, 
Put  in  square  form, 
Take  square  root,       x- 


x^-\-^=2 


x'+^+ 


(1)'=-©' 


Solve  for  x. 


'-I' 


x=  — 


97 
36 
97 
36' 

V97 


.+l=-4 


6 
V97 


.+5=- 


6  ' 
=  .8 


6 
9^ 

6  ' 


97 
36 

V97 
6 

V97 


5 

6  6 
__5    9.8 

6  6 
=  -2.4 


238  BEGINNERS*  ALGEBRA 

EXERCISES 

Solve  by  completing  the  square  and  check: 
1.  a;2+4^=32  2.  n^-2n-l  =  0 

3.  x^-Ax=l  4.  /2-|-22-10/=0 
5.  2x^-Qx+S  =  0  6.  3jc2+6.t;-4  =  0 

7.  36:c2+3  =  36:x;  8.  2x^-9x^-9  =  0 

9.  9a2-9a+2  =  0  10.  9^2_6^._8o  =  0 

11.  (^+2)2  =  4(ic-l)2  12.  2/2  =  7/+ll 

13.  x2-2:x;  =  a2-l  14.  ic2+ 14:^+48  =  0     ' 

15.  4x2+16^+15  =  0  16.  4x2+2jc-6  =  0 

17.  6x2+5x-6  =  0  18.  6x2-5x-^6  =  0 

19.  6x2-29^+35  =  0  20.  3w2-10w-8  =  0 

21.  x2-2x  =  2  22.  2x2+5x-l  =  0 

23.  6w2-19w+10  =  0  24.  oh''+3  =  9h 

180.  An  important  special  case.  Consider  the  quadratic 
equation 

x^~2x+S  =  0 

Solve  by  completing  the  square : 

x^-2x=-S 
^2_2%+l=-2 
(^-1)2= -2 
We  are  required  to  find  the  square  root  of  —2.     But  as 
the  product  of  two  equal  factors  is  never  negative,  but 
always  positive,  no  such  square  root  of  —2  can  be  found. 
No  real  answer  can  be  found  for  the  equation.     If  an  equa- 
tion of  this  kind  should  arise  from  any  problem,  we  should 
say  that  the  problem  is  impossible.     This  can  be  handled 
by  the  invention  of  a  new  number.     The  method  will  not 
be  considered  in  this  book. 

181.  Summary  of  methods  of  solving  a  quadratic  equa- 
tion. The  two  methods  used  fpr  solving  quadratic  equa- 
tions are: 

(a)  Factoring. 

(b)  Square  root  or  completing  the  square. 


SQUARE  ROOTS  AND  QUADRATIC  EQUATIONS    239 

There  is  a  third  method  which  is  not  considered  in  this 
book.  Each  method  has  its  advantages.  In  some  cases 
it  makes  but  little  difference  which  method  is  used.  In 
other  cases  one  of  the  methods  is  greatly  to  be  preferred  to 
the  other.  In  solving  a  quadratic  you  should  select  the 
method  best  adapted  to  the  equation  in  hand. 

Illustration.    3«2—2»  =  0,  factoring,  why? 

x^=  (o  — 1)2,  square  root,  why? 
x^—Sx-\-2  =  0,  factoring,  why? 
2x^-]rSx—3  =  0,  completing  square,  why? 


MISCELLANEOUS   EXERCISES 

Solve  the  following  equations  by  the  method  best  adapted  to  the 

equation  in  hand: 

^ 

1.  Sx^-Ux+S=0 

2.  Sx-^-4:  =  Ux 

3.  3x2-1  =  0 

4.   y2-f3y  =  0' 

5.  2n2=l-4w 

6.  24^  =  /2 

7.  l+:r2-3c=0 

8.  w2+7n+3  =  0 

9.  {'ix-l)x=8 

10.  6/2+2/ =  5 

11.  a2+22(a+5)  =  0 

12.  «24_9o=i9w 

13.  21+R  =  2R^ 

14.  72  =  |[(48+(«-l)(-4)] 

15.  72(7-^2).^  16^2  =  0 

16.  (4+|)2-cl6+2i5:)=0 

17.  16(5-l)2-80(62+66+8)=0 

18.  21(l+6c-3c2)+9(c+l)2  =  0 

19.  ll(ll+5)(ll-5)  =  792 

20.  5(n-l)(«+l)  =  3(3-w)(«+3) 
x{x-l)     x{x+l) 


21. 


22.  (a-i)^=2{a-i) 


4  12 

23.  (a-i)2  =  i  24.  {x-iy-^x'  =  Q 

182.  Graphs.  An  interesting  light  is  thrown  upon  the 
solution  of  quadratic  equations  if  one  considers  the  roots 
of  an  equation  in  connection  with  the  graph  of  the  left- 
hand  side  after  the  equation  has  been  put  in  the  standard 


240 


BEGINNERS'  ALGEBRA 


form.  Verify  the  results  given  below.  State  any  conclu- 
sion you  can  derive  from  them  concerning  the  roots  and 
the  graphs. 


Fig.  63 

(^-1)2  =  0 
1  x  =  h  1 


;2-2a;+3  =  0 
(a;- 1)2= -2 


EXERCISES 

Draw  the  graphs  of  the  following  expressions  and  determine  from 
the  graphs  the  roots  of  the  corresponding  equations  when  each  is 
put  equal  to  zero: 

1.  3c2-2:*;-15  2.  x^-\-2x-S  3.  x^-x-Q 

4.  x^+4:X+S  5.  x^-9  6.  x2-4ic+4 

7.  x+3  8.  (2x-{-Qy  9.  {x-iy 

10.   -ic+4  11.  3a; -5  12.  x^-S 

13.  x^-4x+(j  14.  x2+4x+4  15.  x'--7x+Q 

16.  Show  the  roots  of  ai;^ -3^^+2  =  0  by  drawing  the  graph  of 
x2-3a:4-2. 

183.  Problems.     Solve  the  following  problems : 

1.  The  sum  of  the  squares  of  two  consecutive  integers  is  265. 
What  are  the  numbers? 


SQUARE  ROOTS  AND  QUADRATIC  EQUATIONS    241 

2.  The  sum  of  the  squares  of  two  consecutive  integers  is  150. 
What  are  the  numbers? 

3.  The  sum  of  the  squares  of  three  consecutive  integers  is  50. 
What  are  the  numbers? 

4.  A  tinner  wishes  to  make  a  square  box  3  inches  deep  that  will 
contain  a  cubic  foot.  How  large  a  piece  of  tin  is  needed  if  he  cuts 
a  3-inch  square  out  of  each  corner? 

5.  How  large  a  piece  of  tin  will  be  needed  if  a  square  box  is  to 
be  made  that  will  contain  one-half  a  cubic  foot? 

6.  The  tinner  desires  to  make  a  box  6  inches  longer  than  it  is  wide 
and  4  inches  deep  to  contain  160  cubic  inches  after  cutting  out 
square  corners.  What  will  be  the  dimensions  of  the  piece  from 
which  it  is  made? 

7.  Change  the  capacity  of  the  box  in  the  last  exercise  to  200  cubic 
inches. 

8.  It  is  desired  to  lay  off  a  rectangular  field  that  will  contain  340 
square  rods  if  one  side  is  8  rods  shorter  than  the  other.  Find  the 
dimensions  to  be  used. 

9.  What  must  be  the  dimensions  of  a  rectangular  field  that  is  to 
contain  3  acres  and  have  a  100-rod  fence  around  it? 

10.  After  laying  a  concrete  drive  I  have  enough  concrete  to  lay 
200  square  feet  of  surface.  How  wide  a  walk  can  I  lay  about  a  lily 
pond  20  by  15  feet? 

11.  How  wide  a  strip  must  a  fanner  plow  around  a  field  80  by 
60  rods  to  plow  15  acres? 

12.  A  stockman  stated  that  he  bought  a  number  of  horses  for 
$1200.  Four  died  and  he  sold  the  remainder  for  $10  a  head  more 
than  they  cost,  thereby  making  $200  on  the  transaction.  How 
many  did  he  buy? 

The  area  of  a  circle  (Fig.  54)  is  given  by  the 
formula, 

22 
In  the  following  problems  use  7r=-=-    Remem- 
ber that  this  gives  but  three  significant  figures. 

13.  Find  area  of  a  circle  of  radius  5  inches. 

14.  The  area  of  a  circle  is  100.     What  is  its  diameter? 


242 


BEGINNERS'  ALGEBRA 


15.  Two  circles  are  drawn  with  the  same  center,  one  with  radius 
3  inches  and  one  with  radius  5  inches.  What  is  the  width  of  the 
ring  between  them?  What  is  the  area  of  the  small  circle?  Of  the 
larger?    What  is  the  area  of  the  ring? 

16.  A  button  2  inches  in  diameter  is  to  be  painted  in  two  colors 
as  indicated  in  Fig.  55.  The  area  of  the  ring  is  to  be  the  same  as 
the  area  of  the  inside  circle.  The  area  of  the  in- 
.side  circle  will  be  what  part  of  the  area  of  the  out- 
side circle?  Find  the  radius  of  the  inside  circle; 
also  the  width  of  the  ring. 

17.  How  much  must  be  added  to  the  radius  of 
a  circle  whose  diameter  is  4  inches  to  double  its 
area? 

18.  What  must  be  the  diameter  of  a  quart  cup  if  its  depth  is  to 
be  6  inches?     1  gallon =231  cubic  inches. 

The  volume  of  a  cylinder  of  height  h  and  radius  r  is  irr^. 

19.  A  cylindrical  box  is  to  be  made  to  fit  between  two  shelves 
that  are  8  inches  apart.  The  box  is  to  contain  one  cubic  foot. 
What  must  be  its  diameter? 

Right  triangle  problems: 

The  triangle  in  Fig.  56  has  a  right  angle  at  C  It  is 
called  a  right  triangle  and  the  side  AB  is  called  the  hypote- 
nuse. In  the  figure  the  sides  are 
3,  4,  and  5.  Squares  are  drawn 
on  each  side  and  each  square  is 
divided  into  unit  squares.  When 
the  unit  squares  are  counted  it 
will  be  seen  that 

25  =  9+16 

52  =32+42 


That  is,  the  square  on  the  hy- 
potenuse equals  the  sum  of  the 
squares  on  the  two  other  sides. 


Fig.  m 


SQUARE  ROOTS  AND  QUADRATIC  EQUATIONS    243 


It  has  been  known  for  two  thousand  years  or  more  that 
this  fact  is  true  for  all  right  triangles. 

If  a  and  h  are  the  sides  of  a  triangle  (Fig.  57)  and  c  is  the 
hypotenuse, 

20.  Two  sides  of  a  right  triangle  are  5  and  12.    a 
What  is  the  hypotenuse? 

21.  The  hypotenuse  of  a  right  triangle  is  25,  one 
side  is  7.  The  other  is  how  much  shorter  than  the 
hypotenuse? 


22.  One  side  of  a  right  triangle  is  9,  the  hypotenuse  is  41. 
is  the  other  side? 


What 


23.  Two  sides  of  a  rectangle  are  1  and  2.     What  is  the  diagonal? 

24.  The  hypotenuse  of  a  right  triangle  is  10,  one  side  is  5.     What 
is  the  other  side? 

25.  What  is  the  diagonal  of  a  rectangle  if  the  sides  are  60  inches 
and  11  inches? 

26.  If  the  diagonal  of  a  rectangle  is  26  inches  and  one  side  is  10 
inches,  find  the  other  side. 

27.  What  is  the  diagonal  of  a  square  if  one  side  is  S? 

28.  How  much  shorter  is  the  diagonal  path  across  a  lot  100  by 
175  than  the  sidewalk  along  its  sides? 

29.  How  large  a  square  can  you  cut  from  a 
circular  piece  of  tin  of  one  foot  diameter? 
(See  Fig.  58.) 

30.  Find  the  ratio  of  the  area  of  the  square 
and  the  circle  of  the  last  problem,  giving  two 
decimal  places. 

31.  The  sum  af  the  sides  of  a  right  triangle 
is  35  inches;  the  hypotenuse  is  25  inches. 
Find  the  legs.     Interpret  the  two  answers. 


Fig.  58 


32.  One  side  of  a  right  triangle  is  4  feet  longer  than  the  other. 
Find  the  legs  if  the  hypotenuse  is  20  feet. 


244  BEGINNERS'  ALGEBRA 

33.  The  difference  between  the  sides  of  a  right  triangle  is  5,  the 
hypotenuse  is  45.    What  are  the  sides? 

34.  Two  sides  of  a  rectangle  are  3  and  4.  If  the  shorter  side 
remains  unchanged,  how  much  must  be  added  to  the  longer  side 
to  increase  the  diagonal  by  2? 

35.  Suppose  the  longer  side  of  the  rectangle  in  Ex.  34  were 
left  unchanged.  How  much  would  be  added  to  the  shorter  to 
increase  the  length  of  the  diagonal  by  2? 

36.  If  the  same  amount  were  added  to  both  sides  of  the 
rectangle  in  Ex.  34,  how  much  should  be  added  to  increase 
the  diagonal  by  2? 

37.  The  height  of  a  certain  flagstaff  is  unknown.  But  it  is 
noticed  that  a  flag  rope  fastened  at  the  top  is  4  feet  longer  than  the 
pole  and  that  when  stretched  tight  the  rope  reaches  the  ground  20 
feet  from  the  base  of  the  pole.    How  high  is  the  pole? 

38.  A  flagstaff,  AB,  50  feet  high,  was  broken  off  at  a  point  C, 
the  broken  end  rested  on  C  and  the  top  touched  the  ground  at  D, 
30  feet  from  the  base  of  the  staff.  Find  the  length  of  the  part  still 
standing. 


CHAPTER  XI 

Fractional  Equations  in  One  Unknown 

184.  Definition.  Equations  frequently  occur  in  which 
the  unknown  appears  in  the  denominator  of  a  fraction. 
Such  equations  are  called  fractional  equations. 

1=1-4-1        5     ^3 

are  fractional  equations. 

185.  Solution.  A  fractional  equation  is  usually  best 
solved  by  multiplying  both  sides  by  an  expression  that 
will  remove  all  the  denominators  of  the  fractions.  The 
least  common  denominator  should  be  used  for  this  ptupose. 
Why? 

This  is  called  clearing  the  equation  of  fractions. 

The  resulting  equation  may  be  a  linear  equation,  a  quad- 
ratic equation,  or  an  equation  of  still  higher  degree. 

All  the  roots  of  the  fractional  equation  will  be  roots  of 
the  new  integral  equation.  But  the  reverse  statement,  that 
all  the  roots  of  the  new  equation  are  roots  of  the  fractional 
equation,  is  not  necessarily  true.  Some  new  roots  might 
be  introduced  by  the  multiplication.  In  consequence  of  this 
fact  it  is  necessary  to  test  all  roots  found  from  the  integral 
equation  by  substituting  them  in  the  original  fractional 
equation. 


Illustration  1: 

3  ,  .     1  Check:  3   ,  . 

Solve  -+5  =  -  -2+5 

Multiply  both  sides  by  2x,  3 

6H-10:c  =  ^  9,  - 

9.v=  -6  "2"^^ 

.3  2 

245; 


246 


BEGINNERS'  ALGEBRA 


Illustration  2: 
4 


Solve 


1^.= 


1 


=  0 


x-Z      x-\-Z      a;2-9 
Multiply  both  sides  by  x^  -9, 

4(a:+3)-(x-l)(a;~3)+a;2-l  =  0 

Ax-^l2-{x''-^x-\-Z)+x^-l  =  0 

4:x;+12-:»;2^4:c-3+ic2-l  =  0 

8:^4-8  =  0 

x=-l 


1-11-1 


Check:     -1-3     -1+3^1-9 
-1+1+0 

Note.     Clearing  of  fractions  applies  to  equations  only  and  is  not 
allowed  in  checking. 


EXERCISE   I 


Apply  to  the  following : 
3.  7-i=4 

X 

5,  ?H-6  =  l-:r 

X 


2. 


1     1     2 


4.  4+5=« 

X  X 


The  essential  difference  in  the  character  of  a  fractional  and 
an  integral  equation  can  best  be  shown  by  the  pictures  or 
graphs  of  the  left-hand  number  when  all  terms  are  put  on 
that  side.  For  instance,  take  the  equations  of  Illustration 
1  just  given.     (See  Fig.  59.) 


3    9 
Put  the  fractional  equation  in  the  form   -+^  =  0 


The  integral  equation, 


(1) 

6+9«  =  0  (2) 


-5 

-4 

-3 

-2 

-1 

—  .5 

0 

1 

2 

3 

4 

5 

a)M 

3.9 

3.8 

3.5 

3 

1.5 

-1.5 

7.5 

6 

5.5 
33 

5.3  5.1 

(2)  6+9x- 

-39 

-30 

-21 

-12 

-3 

1.5 

6 

15 

24 

42 

51 

FRACTIONAL  EQUATIONS   IN  ONE  UNKNOWN    247 

The  point  where  the  graph  cuts  the  horizontal  axis  indi- 
cates the  root  of  the  equation.     The  two  graphs  cut  the 


« 


Fig.  59 

^-axis  at  the  same  point,    —  f,  which  is  the  root  of  each 
equation. 

Let  the  student  draw  in  a  similar  way  the  graphs  for 
Exercise  5,  below. 

EXERCISE   II 

Solve: 

2     1,.  ^    1.1     2 

5     5  4    6     0; 

3.  i^+^  =  ^  4.  -+-  +  1=0 

40    d     10  mm 

^-  *+"    j:  **•     4        «-3 


248  BEGINNERS'  ALGEBRA 


7.  I       6^0  8.^=7 

5    x—l  x—1 

^   i_l  ,^    x+1    x+7 


10. 


"•  2(1-/)     4  ^^'  x-1     a;+3 

11    ^^+^z:2^_40_  _6 y+1       y    _Q 

•^-  a:-2^a;+2     a;2-4  ^^-  3;+2     3,-2  V-4~ 
iQ    ^+3    x-l_7x-4:  x-2  _4^-16     1 

^^-  ^-1     x+1     x'-l  ^^-  x^-^x     x^-Sx'^x 

15-37-3-^1  1^-5+5^^:^=3 

17      ^    _3_L  14  w-2_n4-5  ■    14-15» 


a;-6  x+6  w+3    w-4    W2-W-12 

n— 1       ^— 2_^+l  ^pj    a;— 5a_l     5a 

^'  2n-6    3w-9~    6  ^*     ba    ~2    2x 

21.  ^-1  =  ^ 
a  X 

23.  55_?LZ55=25 
o;         b 

a  b      , 

25.  T — a  =  —  -b 

bx  ax 

27.  ^-'i^+b=0 
X         5x 

m    ,     n  2m 

29.  —, h 


31. 


x-\-n    x-\-m     x+m 
a      .      b  2bH 


22. 

0—6     a 
X         b 

?4 

x—a     1     aa;  — 1 

ax       a        X 

26. 

1      6  _:x;4-2a     a 
b     ax       ax       bx 

28. 

x—a .x—b      1 
bH       aH      ab 

30. 

1        a-6_^    1        a+6 
a-6        ic       a+fr        « 

6ic  — 1     a:K  — 1      {ax  —  l){bx  —  l) 


186.  Transformation  of  formulas.  In  dealing  with  for- 
mulas that  are  solved  for  one  letter  it  is  often  desirable  to 
derive  a  formula  solved  for  some  other  letter.  When  one 
is  solving  for  any  letter,  all  other  letters  are  to  be  considered 
as  known.  Attention  should  be  fixed  on  the  letter  for  which 
the  formula  is  to  be  solved,  and  the  best  methods  of  untan- 
gling it  from  the  other  numbers  should  be  determined  step 
by  step. 


FRACTIONAL  EQUATIONS  IN   ONE  UNKNOWN    249 

Illustration  1: 

n-\-i 
Solve  for  w,  ^  =  r— — . 

Clear  of  fractions,  k{l-\-i)=n-\-i 

Subtract  i  from  both  sides,      k{l+i)  —i  =  n 

It  may  be  left  in  this  form  or  expanded,  as  you  wish, 

n  =  k-\-ki—i 
Illustration  2: 

Solve  the  same  formula  for  i. 
Clear  of  fractions,  k{l-\-i)  =  n-\-i 

Expand,  collect  terms,  and  divide  by  the  coefficient  of  i. 
k-\-ki=n-\-i 
ki—i  =  n—k 
{k—l)i=n—k 

.    n—k  ' 

You  may  test  the  correctness  of  your  work  by  substituting  your 
resiilt  in  the  original  formula.  Show  how  each  line  is  obtained 
from  the  one  above  it. 

,  n—k 


_  (k  —  l)n-{-n—i 
k-l+n-k 
,_kn—n-\-n—k 

~  k-l-\-n-k 
,     kn  —  k 

k=k 


EXERCISES 

Transform  the  formulas  as  indicated: 

1.  m=  — .     Solve  for  n.  2.  - —  =  m.    Solve  for  y. 

n  X 

17 


250  BEGINNERS*  ALGEBRA 

m—3 


3.  -+-^  =  1.    Solve  for  y. 
a      0 

5.  l  =  7-r — .    Solve  form. 
l-\-m7i 

7.  r=- .     Solve  ford. 

9.  S  =  -^.     Solve  for /,r. 
r  —  l 


4.  1  = 


►.  r= 


l+3m 
2 


1+/ 


Solve  for  m. 
Solve  for  „. 


8.  r=^.     Solveiorp. 
l-\-e 

10.  F  =  ^^^~}\     Solve  for  i2.  . 

A+1 


11.  n  =  l(^—^).     Solve  for  ^.       12.  R=-^.     Solve  for  r,  n. 

13.  T7--T^  =  7^-     Solve  for   each   letter   and   find   M  when 
M     t,     ^ 

£  =  36^.25,  5  =  29.53. 

14.  a  =  /> r.     Solve  for  w. 

n  —  1 


15.  C  = 


^+^ 


-.     Solve  for  E,  7?,  r,  iV. 


T     T 

16*  S  ~Ti-  = g'     Solve  for  each  letter. 

Mm 

1^-  ^=wS^-    Solve  fori?. 

18.  5=^+^.     Solve  for  P,  for  C. 

19.  Pf=-^..     Solve  for/. 

20.  y^Tl  9L+^  )•     Solve  for  each  letter. 


21.   /: 


En 

'^+R 
m 


.    Solve  for  R,  for  n. 


187.  Problems.  Several  types  of  problems  lead  natu- 
rally to  fractional  equations.  A  few  illustrations  are  given 
in  this  article. 

1.  If  36  is  divided  by  a  certain  number,  the  quotient  is  5  more 
than  the  number.    What  is  the  number? 


FRACTIONAL  EQUATIONS   IN   ONE   UNKNOWN    251 

2.  The  quotient  of  8  divided  by  a  certain  number  exceeds  9 
times  the  nimiber  by  6.     What  is  the  nimiber? 

3.  Divide  60  into  2  parts  such  that  their  quotient  is  equal  to  f . 

4.  Divide  10  into  2  parts  so  that  their  ratio  is  twice  one  of  the 
parts. 

5.  What  number  must  be  added  to  both  numerator  and  denomi- 
nator of  Yx  in  order  that  the  resulting  fraction  shall  equal  f? 

6.  What  number  added  to  both  numerator  and  denominator  oi 
the  fraction  f  will  double  the  value  of  the  fraction?  What  number 
will  halve  the  value  of  the  fraction? 

7.  What  number  must  be  added  to  the  denominator  of  j^  in 
order  that  the  resulting  fraction  shall  be  equal  to  ^? 

8.  What  number  must  be  added  to  the  denominator  of  ^f  in 
order  to  get  a  fraction  equal  to  -f-  ? 

9.  What  ntunber  added  to  both  numerator  and  denominator  of 
the  fraction  f  will  double  the  value  of  the  fraction? 


Triangles  of  the  same  shape  are  called  similar  triangles. 

The  right  triangles  A  and  B  are  similar,  as  are  also  C  and 
D  (Fig.  60).  It  is  shown  in  geometry  that  the  ratios  of  cor- 
responding sides  of  two  similar  triangles  are  equal,  that  is, 


b~b' 
Verify  this  relation  by  measuring  the  sides  in  Fig.  60. 


252 


BEGINNERS'  ALGEBRA 


10.  In  two  similar  triangles  if  a  =  7,  6=12,  and  6' =  36,  what 
will  a!  equal? 

Ifa  =  5,  a'  =  12,  and^>'  =  40,  6  =  ?  , 
If  6  =  7,  a=15,  anda'  =  9,  y=? 
If  0  =  25,  o'  =  40,  and  Z>'  =  9,  6=? 

This  relation  between  the  sides  of  similar  triangles  may 
be  used  to  estimate  the  height  of  a  building  or  tree. 

11.  Suppose  a  spire  casts  a  shadow 
of  60  feet  while  at  the  same  time  the 
shadow  of  a  6-foot  post  is  4  feet  long. 
Find  the  height  of  the  spire. 

60     4 
6-60 


We  have  ^^  =  -r 


whence 


h  = 


90 


B 


12.  It  is  desired  to  find  the  height 
of  a  flag  pole  on  a  cloudy  day. 

\A\,  ^5  be  the  flag  pole;  CD  be  a  yardstick  or  any  rod;  and  E 
the  eye  of  a  boy  lying  on  the  ground  so  that  he  can  just  see  the  top 
of  the  flag  pole  over  the  end  of  the 
pole  CD  (Fig.  62).  What  lines  must 
be  measured  to  find  the  height  of  the 
flag  pole?  Using  x  for  the  height  of 
the  flag  pole  and  a,  6,  c  for  the  meas- 
ured lengths,  state  a  formula  for  x. 
*  Or,  if  the  measuring  rod  is  long 
enough,  the  pole  and  rod  can  be 

sighted  in,  while  the  boy  is  standing  as  in  Fig.  63.     Explain  how  the 
height  oi  AB  will  be  found  then. 

Assign  values  to  these  measured 
lines  and  solve  for  the  other. 


^-.D 


Fig.  62 


B 


What  is  the  height  of  a  tree  when 
it  was  measured  in  this  last  way  by 
a  boy  whose  eye  was  5  feet  from  the 
ground,  an  8-foot  pole  being  placed 
6  feet  from  the  boy?  The  distance  from  the  boy  to  the  tree  was 
found  to  be  320  feet. 


Fig.  63 


FRACTIONAL  EQUATIONS   IN   ONE    UNKNOWN     253 

13.  How  far  away  is  a  tower  which  is  known  to  be  150  feet  high, 
if  its  top  can  just  be  seen  over  an  8-foot  pole  placed  20  feet  from  a 
man  with  his  eye  at  the  ground. 

14.  In  Fig.  64  it  is  desired  to  find  the  distance  AB.  It  is 
impossible  to  measure  the  dis-  j) 
tance  directly  because  of  the  ^T"*^ — 
river  between.  AC,  AE,  smdCD  ^  q 
were  measured.  ^C  =  200,  AE  _^,^.^ 
=  100,  CD  =125.  How  wide  is  C  >s 
the  river?  '        p^^   ^^^ 

The  following  are  work  problems: 

15.  A  pipe  delivers  water  to  a  150-gallon  tank  at  the  rate  of 
20  gallons  a  minute.  How  long  will  it  take  to  fill  the  empty  tank? 
If  a  pipe  can  fill  this  tank  in  10  minutes,  how  many  gallons  does  it 
supply  in  one  minute?    What  part  of  the  tank  will  it  fill  in  one 

•minute?     In  5  minutes? 

16.  A  pipe  can  fill  a  certain  tank  in  12  minutes.  What  part  of 
the  tank  can  it  fill  in  one  minute?    In  7  minutes? 

17.  Smith  can  do  a  piece  of  work  in  3  days.  What  part  of  the 
work  can  he  do  in  one  day?  In  2  days?  In  3  days?  Jones  can 
do  a  piece  of  work  in  x  days.  How  much  of  the  work  can  he  do  in 
one  day?    In  4  days?     In  n  days? 

18.  If  one  pipe  will  fill  a  cistern  in  5  hours,  what  part  of  the 
cistern  can  it  fill  in  1  hour?  If  a  second  pipe  can  fill  it  in  9  hours, 
what  part  of  the  cistern  can  be  filled  in  1  hour  if  both  pipes  are 
open  at  the  same  time? 

19.  A  tank  containing  240  gallons  is  supplied  by  two  pipes,  one 
delivering  2  gallons  a  minute  and  the  other  3  gallons  a  minute. 
How  long  will  it  take  to  fill  the  tank  when  both  pipes  are  turned  on? 

The  equality  to  be  used  here  is: 

Total'  number  of)        (total  number  de- 1        (  total  num- 
gallons  delivered  f  ~^  \  livered  by  second  f  —  \     ber  tank 
by  first  pipe    J        (  pipe  J        (^can  contain 

If  ^=the  unknown  number  of  minutes,  the  equation  is 

2/+3/=240 
Therefore  ^=48 


254  BEGINNERS'  ALGEBRA 

20.  Suppose  a  tank  of  240  gallons  can  be  filled  by  one  of  two 
pipes  in  20  minutes  and  by  the  other  in  30  minutes.  How  long 
will  it  take  to  fill  the  tank  when  both  pipes  are  turned  on? 

The  same  equality  that  was  used  in  Exercise  5  can  be  used  here: 

Total  number  of )        ( total  number  de- )        (  total  num- 
gallons  delivered  >  +  <livered  by  seconds  =  <^  ber  tank  will 
by  first  pipe     J        (^  pipe  J        (^       hold 

Let  /=the  unknown  number  of  minutes  to -fill  the  tank 

240 

^  =  number  of  gallons  first  pipe  delivers  in  one  minute 

240 

^  =  number  of  gallons  second  pipe  delivers  in  one  minute 

240      240 
Hence  the  equation  is  -27:^+-o?r^=240 

Such  work  problems  are  often  solved  by  the  use  of  another  equal- 
ity, namely: 

Part  of  tank  filled  |        j  part  of  tank  filled  j        f  part  of  tank  filled  in 
by  first  pipe  in   >  +  <  by  second  pipe  in  >  =  <  one  minute  by  both 
one  minute      J        (^      one  minute      J        (  working  together 

Here 

— =the  part  of  tank  filled  by  first  pipe  in  one  minute 

—  =  the  part  of  tank  filled  by  second  pipe  in  one  minute 
oO 

-  =  the  part  of  tank  filled  by  both  pipes  in  one  minute 

Hence  _L4-_L    1. 

20"^30''  / 

Solve  for  /. 

It  will  be  noticed  that  this  solution  takes  no  account  of  the 
capacity  of  the  tank.  In  the  first  solution  the  capacity  of  the  tank 
was  used,  but  had  no  effect  in  determining  the  unknown  time,  for 
the  240  was  divided  out  of  the  equation. 

21.  One  pipe  can  fill  a  cistern  in  42  hours;  another  can  fill  the 
cistern  in  14  hours.  How  long  will  it  take  to  fill  the  cistern  when 
both  pipes  are  turned  on? 


FRACTIONAL  EQUATIONS  IN  ONE  UNKNOWN    255 

22.  One  pipe  can  fill  a  cistern  in  2  hours;  a  pump  can  empty  it 
in  5  hours.  How  long  will  it  take  to  fi.ll  the  cistern  if  both  supply- 
pipe  and  pump  are  working  at  the  same  time? 

23.  A  tank  of  a  certain  capacity  can  be  filled  from  one  pipe  in 
30  minutes,  from  another  in  60  minutes,  and  can  be  emptied  by  a 
pump  in  40  minutes.  How  long  will  it  take  to  fill  the  cistern  when 
all  three  are  working? 

24.  A  cistern  can  be  filled  in  12  minutes  by  two  pipes.  One  of 
them  will  fill  it  in  20  minutes.     In  what  time  would  the  other  fill  it? 

25.  A  can  do  a  piece  of  work  in  m  hours  and  B  in  n  hours.  How 
long  will  it  take  them  to  complete  the  work  if  they  work  at  the  same 
time? 

26.  What  will  be  the  answer  to  Exercise  25  if  (a)  w=3,  »=  15; 
(6)  w=15,  n  =  5;  {c)  m  =  7,  »  =  5. 

27.  One  man  takes  4  hours  longer  than  another  to  saw  a  cord  of 
wood  and  both  working  together  can  saw  it  in  3  hours.  How  long 
does  it  take  each  to  saw  a  cord? 

28.  A  tank  can  be  filled  in  3  hours  by  two  faucets  flowing  at 
the  same  time.  It  will  take  one  failcet  8  hours  longer  to  fill  it 
than  it  will  the  other.  In  what  time  can  the  tank  be  filled  by 
each? 

29.  A  certain  number  of  letters  are  to  be  addressed.  A  can 
address  them  in  10  hours.  B  can  do  the  work  in  15  hours.  How 
long  will  it  take  them  to  address  the  letters  if  they  work  at  the 
same  time? 

30.  A  can  do  a  piece  of  work  in  6  days.  It  takes  B  three  times 
as  long  to  do  the  same  work.  How  long  will  it  take  them  to  do  the 
work  when  they  work  at  the  same  time? 

31.  When  working  at  the  same  time  A  and  B  can  do  a  certain 
piece  of  work  in  2  days.  A  when  working  alone  can  do  it  in  3  days. 
How  long  would  it  take  B  to  do  the  work? 

32.  A  and  B  can  do  a  piece  of  work  in  2  days.  When  working 
alone  it  will  take  A  3  days  longer  than  B  to  do  the  work.  How 
long  will  it  take  each? 


256  BEGINNERS'  ALGEBRA 

33.  A  tank  can  be  filled  by  one  pipe  in  6  hours,  by  another 
in  9  hours,  and  by  a  third  in  18  hours.  How  long  will  it  take  to 
fill  the  tank  when  all  are  running? 

34.  A  tank  can  be  filled  by  three  pipes  in  6  hours,  by  one  of  them 
in  12  hours,  by  another  in  36  hours.  How  long  would  it  take  the 
third  one  to  fill  the  tank? 

The  following  are  motion  problems: 

35.  A  man  who  can  row  in  still  water  at  the  rate  of  r  miles  an 
hour  is  rowing  in  a  river  whose  current  is  2  miles  an  hour.  How 
fast  can  he  go  upstream?  Downstream?  How  long  would  it  take 
him  to  row  5  miles  upstream?    6  miles  downstream? 

36.  A  boatman  rowed  7  miles  upstream  and  back  in  8  hours. 
If  the  speed  of  the  current  was  3  miles  an  hour,  what  was  his 
rowing  rate  in  still  water? 

37.  How  far  can  a  person  who  has  8  hours  to  spare  ride  in  a  wagon 
at  6  miles  an  hour  so  that  he  can  return  walking  at  the  rate  of 
4  miles  an  hour  and  arrive  home  on  time? 

38.  A  certain  river  boat  that  can  travel  at  the  rate  of  9  miles  an 
hour  takes  5  hours  and  30  minutes  to  make  a  trip  of  20  miles  up 
the  river  and  back,  allowing  a  stop  of  30  minutes.  What  is  the 
average  rate  of  the  current? 

39.  Another  boat  made  the  same  trip  in  4  hours  and  30  minutes. 
What  was  its  rate  of  travel? 

40.  The  boat  of  Exercise  38  is  to  take  a  pleasure  party  up  the 
river  to  be  gone  just  2  hours.  How  soon  after  starting  must  the 
captain  turn  back?    How  far  up  can  he  go? 

41.  How  much  time  should  the  captain  of  the  first  boat  allow 
for  a  trip  of  15  miles  up  the  river  and  back  with  a  stop  of  one 
hour? 


CHAPTER   XII 

Sets  of  Equations  in  Two  Unknowns 

188.  Introduction.  In  chapter  vi  we  learned  that  it  is 
often  simpler  to  solve  a  problem  by  using  two  unknowns 
instead  of  but  one.  It  was  noticed  that  the  use  of  two 
unknowns  required  the  use  of  two  equations  and  that 
these  two  equations  must  be  furnished  by  the  conditions  of 
the  problem. 

For  instance,  the  sum  of  two  numbers  is  27;  their  difference  is  5. 
What  are  the  numbers? 

x—y  =  5 

In  the  problems  we  have  considered  thus  far  both  equa- 
tions of  the  pair  or  set  were  of  the  first  degree.  But  many 
problems  give  equations  of  a  higher  degree  than  the  first, 
and  others  require  fractional  equations. 

For  instance,  the  sum  of  the  squares  of  two  numbers  is  34;  the 
difference  between  the  numbers  is  2.  What  are  the  numbers? 
The  equations  are 

x-y  =  2 

The  study  of  all  the  different  kinds  of  sets  that  may  arise 
is  beyond  the  scope  of,  this  book.  We  will  consider  only  a 
few  of  the  simpler  types,  namely:  (1)  both  equations  of  the 
first  degree;  (2)  one  equation  of  the  first  degree  and  one  of 
the  second;  (3)  either  or  both  equations  fractional. 

189.  Both  equations  of  the  first  degree.  This  type  has 
been  considered  in  chapter  vi.  Describe  the  three  methods 
used  in  solving  a  set  of  two  linear  equations:  by  graphs,  by 
addition,  and  by  substitution. 

257 


258         /  BEGINNERS'  ALGEBRA 

190.  Inconsistent  equations.  If  an  attempt  is  made  to 
solve  the  two  equations 

by  subtraction,  we  get  the  statement 

.     0  =  3 
which  is  absurd. 

The  two  equations  are  inconsistent;  that  is,  they  cannot 
both  be  true  for  the  same  pair  of  values  of  x  and  y.  Draw 
the  graphs  of  these  two  equations  and  state  what  is  their 
relation  to  each  other. 

191.  Dependent  equations.  In  the  attempt  to  eHminate 
one  of  the  unknowns  from  the  set 

2x+Sy  =  5 
4:^4-6^;  =  10 
all  the  terms  disappear. 

What  is  shown  when  the  graphs  of  these  two  equations  are 
drawn  ?  One  of  these  is  really  dependent  upon  the  other  and 
the  equations  are  equivalent,  and  any  solution  of  one  will  be 
a  solution  of  the  other.  They  determine  no  definite  solution. 
Two  equations  are  independent  if  one  cannot  be  derived 
from  the  other. 

192.  Solution.  It  follows  from  the  preceding  articles 
that  a  definite  solution  can  be  found  for  a  set  of  two  linear 
equations  in  two  unknowns  only  when  the  equations  are 
consistent  and  independent. 

What  can  you  say  of  the  equations  if  their  graphs  are 
parallel  or  if  they  coincide  ? 

EXERCISES 

In  the  following  problems  use  the  method  which  seems  to  be  best 
adapted  to  the  equations  at  hand: 

1.  4,x-\-5y  =  ^0  2.  20  =  7p-\'2q 

6a;-7y=2  55  =  20p-\-7q 


SETS  OP  EQUATIONS  IN  TWO  UNKNOWNS      259 


3.  3^=6  4.  5-\-a-\-2b  =  0 

5^+5  =  5  7+5a-\-b  =  0 

5.  The  sum  of  two  numbers  is  200;  their  difference  is  79.  What 
are  the  numbers? 

6.  In  what  proportions  should  two  kinds  of  coffee  worth  20  cents 
and  30  cents  a  pound  be  combined  to  make  a  mixture  of  50  pounds 
worth  26  cents  a  pound? 


7.  4^^+3^  =  6000  8.  5.4+3B=19 

ic-2>'  =  2000  2A-  B=l 

9.  5>'  =  3-x-|-3v 
2x+3  =  9-4>' 


10.  25  =  5 


m 


/=a+(5-l)2 


12.  a+l<l^l 


11.  %-l-R  =  0 
4       2 

13.  4{x-3y)  =  S 

(x+y)=S(x-2y) 

15.  9  =  k-k' 

9  =  19ife-36U' 

1V.3(M)  =  5,    4-3(|+|)  =  2 

18.  The  sirni  of  two  numbers  is  s,  and  their  difference  i?  i.     Find 
the  numbers. 

In  the  following  four  examples  solve  for  x  and  y: 


14.  5{x-2y)-{x-y)=-24: 
n(2x+3y)-\-(2x-y)  =  200 

16.  3:c+4y  =  25 

3(3+^)+4(4+y)=0 


19.  2x+y=2a 

20.  ax=by 

2x-y=2b 

x-{-y=c 

21.  ax+by=l 
bx+ay=l 

22   ""  ■+■  y 

a-{-b^a-b 
x-y_ 
4ab 

23.  Find  a  and  b 

in  terms  ol 

:  X  and  y: 

x  = 

Sa+2b 

y= 

a-{-5b 

2a 


260  BEGINNERS'  ALGEBRA 

24.  Solve  for  p  and  n: 

180+(2/>+2w)  =  i2 
l^-{2p-2n)  =  R' 

193.  Fractional  equations.  In  some  instances  fractional 
equations  in  two  unknowns  can  be  reduced  to  linear  equa- 
tions by  clearing  of  fractions.     The  solution  easily  follows. 

3%+4>'  =  25  (1) 

-A_  =  _^  (2) 

4+3;     3+:^  ^  ^ 

Clearing  of  fractions,  3(3+^)  =4(4+7)  (3) 

Let  the  pupil  complete  the  solution. 

EXERCISES 

Solve  for  x  and  y: 

1.  i-f=:^  2.  ^f^^i  3.  3;c+4y=25 

3    4     12  x-\-y-^    2  4  3 

^=1  ^=5  ^"^H=^ 

y-Z  y 

A    ^-3    4                    _    %-y-\-\     _                    r,    ic+1     1 
4.  -  =  -  5. — r  =  5  b.  =  - 

y  —  \     5  X— y  — 1  >>       a 

x-y_x     1  x+y+l_g  3g    _1 

3        6     2  :i;-fy-l  y+1     6 

7.^^=6  8.  ^^^  =  i  9.^^=1 

y  •  >'        3  X 

X    _  :y+y— 3_1  ac— y_g 

y-\-a  X  ^  y 

194.  More  complicated  fractional  equations.  If  an  at- 
tempt is  made  to  solve  a  set  like 

-+-  =  5  (1) 

X     y 

?-i  =  3  (2) 

•      X     y 

by  clearing  of  fractions,  we  get 

y-\-Zx  =  bxy 
2y—x  =  Sxy 


SETS  OF  EQUATIONS  IN   TWO  UNKNOWNS      261 


The  resulting  set  is  much  too  difficult  for  us  to  solve  in  this 
course.  Do  not  clear  such  sets  of  fractions.  Simply  elimi- 
nate one  of  the  fractions  containing  an  unknown  just  as 
in  linear  sets. 

1        o 

(1) 

(3) 


Thus, 

X     y 

Multiply  (2)  by  3, 

5-3  =  9 

X     y 

Add  the  members  of  the  equations 

t 

1  =  14 

X 

l  =  Ux 

1 
2  =  ^ 

Substitute  in  (1), 

2+?  =  5 

y 

2). 

Whence, 

y  =  l 

Check  in  (1)  and  ( 

EXERCISES 

Solve  for  x  and  y: 

1.  1+1=6 

-hh' 

3. 

H- 

?-l  =  3 

X    y 

H- 

^-1- 

^■H-- 

>  M=.« 

6. 

1-1-1 

H'-  ■ 

X     y 

1-1=* 

X    y 

-  \^A 

'H-^ 

9. 

3     a 
-+-  =  w 
X     y 

1_1_1 

1         1             ^ 

2_b__ 

X    y     \} 

Sx     2y~      ^ 

X   .y    ^ 

10.  "-'+^=1 
x^y 

11.^+^^=1 

X    y 

12. 

a  ,  b 
X     y 

x^y 

»-*  =  5 
X     y 

X    y 

262  BEGINNERS'   ALGEBRA 

195.  One  equation  of  the  first  degree,  one  of  the  second. 

As  an  illustration  take  the  set 

2:^+^  =  10  (1) 

^24.^2  =  25  (2) 

The  most  obvious  plan  to  follow  in  solving  this  set  is  to 
eliminate  one  of  the  unknowns  by  substitution.  Why  not 
by  addition? 

It  makes  no  real  difference  which  unknown  is  chosen  to 
be  eliminated.  In  this  case  y  is  the  better  one  to  choose. 
Can  you  tell  why?  From  the  linear  equation  obtain  y  in 
terms  of  x. 

y  =  l0-2x  (3) 

Substitute  this  value  of  y  in  place  of  y  in  the  quadratic 
equation  r^2_{_(io-2^)2  =  25  (4) 

Expand  and  solve, 

a;2+100-40A;+4^2==25 

5x2-40^+75  =0 

ii:2-8%+15  =  0 

(ic-3)(x-5)=0 

x  =  3,  x  =  5 

It  will  be  noticed  that  in  the  solution  we  replaced  equa- 
tion (2)  by  equation  (4) ;  that  is,  to  solve  the  set  (1)  and  (2), 
we  solve  the  set 

2x-\-y=l0  (1) 

x'-{-il0-2xy=^25  (4) 

Consequently,  to  get  the  values  of  y  that  correspond  to 
the  values  x  =  S  and  x  =  5,  we  must  substitute  these  values 
in  succession  in  equation  (1),  the  linear  equation,  or,  what 
is  the  same  thing,  in  (3). 

Thus,  ioT     x  =  3  and  for       x=^6 

y=10-Q  :)/  =  10-10 

=  4  =0 


SETS  OF  EQUATIONS   IN  TWO  UNKNOWNS      263 

there  are  two  solutions  to  the  set  of  equations,  each  com- 
posed of  a  pair  of  values,  one  for  each  unknown. 
Check  by  substituting  in  both  equations. 

It  is  very  important  that  the  right  values  be  paired. 

:x:  13  15 


or, 


4  10 


It  is  perfectly  evident  that  the  same  method  of  solution 
may  be  applied  to  any  set  of  equations  of  this  kind.  The 
method  may  be  stated  in  the  form  of  a  rule : 

Rule.  (1)  Using  the  linear  equation,  express  one  un- 
known in  terms  of  the  other. 

(2)  Eliminate  this  unknown  by  substituting  its  value  in 
the  equation  of  second  degree. 

(3)  Solve  the  resulting  equation  in  one  unknown. 

(4)  Find  the  values  of  the  unknown  that  was  eliminated 
by  substituting  the  values  found  for  the  other  unknown  in 
the  linear  equation,  keeping  the  pairs  of  corresponding 
values  together. 

(5)  Check  by  substituting  in  both  of  the  original  equations. 

EXERCISES 


Solve : 

I.  2x-\-y  =  0 
>2=16:c 

2.  y  =  x^-6x+S 
y  =  2x+8 

3.  2y-x  =  5 
:r2+y2  =  25 

4.  x-y  =  S 
xy=^0 

5.  y'  =  Ax 
2x-^y=4: 

6.  y-h5x  =  x^-\-4: 
Sx-\-y=4: 

7.  y-2x^  =  5- 

2x  =  4:-y 

-5x 

8.  y-\-5x  =  x^+4: 
y  =  x—5 

9.  a;2+3;2  =  49 
5x-4y  =  0 

196.  Graphs.  An  interesting  Hgh't  may  be  thrown  upon 
the  solution  of  sets  of  equations  of  the  kind  we  have  just 
been  considering  by  the  following  means:  Draw  the  graphs 
of  the  two  equations  of  the  set  on  the  same  axis  just  as  we 
did  with  a  set  of  two  linear  equations,  and  then  note  their 


264 


BEGINNERS'  ALGEBRA 


points  of  intersection.     For  illustration  take  set  5  of  the 
last  article: 

y^  =  4tx  (1) 

2x-\-y=^4:  (2) 

Read  off  the  coordinates  of  the  inter- 
section points  of  the  two  graphs  (Fig.  65). 
Show  that  the  points  of  the  intersection 
of  the  two  graphs  correspond  to  the 
common  solutions  of  the  two  equations 


the  set. 

Fig.  65 

The  following 

three  sets  of  equations  are  treated  in  the 

me  way: 

(a) 

(b)                                 (c) 

y=x^-5x+4: 

y  =  x^  —  5x-\-4:            y  =  x^  —  5x-\-4: 

y  =  x  —  4: 

y=x—5                     y=x—Q 

Graphs 


(a) 

X 

4 

2 

y 

0 

-2 

Fig.  66 
Solutions 


(c) 
No  solution 


In  (a)  the  graphs  intersect;  there  are  two  different  solu- 
tions. 


X 

(b) 
3 

3 

y 

-2 

-2 

SETS  OF  EQUATIONS   IN   TWO  UNKNOWNS      265 


In  (b)  the  graphs  touch  each  other;  the  two  solutions  are 
equal. 

In  (c)  the  graphs  do  not  intersect;  there  are  no  real 
solutions. 

197.  Graphs  of  equations  of  the  second  degree.    The 

graph  of  an  equation  of  the  first  degree  is  always  a  straight 
line.  But  equations  of  the  second  degree  have  graphs  of 
various  kinds.  It  wnll  be  interesting  to  notice  a  few  of  them. 
In  computing  the  table  of  pairs  of  values  to  be  plotted  it 
will  be  found  most  convenient  to  alter  the  form  of  the  equa- 
tion so  that  one  letter  is  expressed  in  terms  of  the  other. 

Thus,  Sx-y-hx'-  =  5 

becomes  y  =  x^-\-3x  —  5 

The  equation  X"+y^  =  25  is  a  little  more  difficult  to  handle. 

It  becomes 

f~  =  25-x^ 
y  =  ±yl25-x'^ 
For  every  value  given  to  x  there  will  be  two  values  for  y. 


y=-{-^l25-x^ 

-^25-x' 

X               y 

y 

0     V25     =5 

-V25     =-5 

1     V24     =4.9 

-V24     =-4.9 

2     V2r     =4.6 

-V21      =-4.6 

3     Vl6     =4 

-VT6     =-4 

4     V9       =3 

-V9       =-3 

5     Vo       =0 

-Vo      =0 

6     V-ll  = 

-V-ll  = 

i'- 


m 


m 


Fig.  67 

What  are  the  values  of  y  when  x  is  given  negative  values  ? 
Make  such  a  table  and  plot  all  results.  Notice  that  the 
point  for  :j£;  =  6  cannot  be  plotted.     Why? 

Be  sure  to  take  a  sufficient  number  of  points  to  determine 
the  position  of  the  graph  fairly  well. 

18 


1200 


BEGINNERS'  ALGEBRA 


EXERCISE    I 

Plot: 

1.  y-x^=4  2.  y^  =  ^ix 

3.  x^+y^=lG  4.  a:-^+/  =  30 

5.  :^2_3,2=36  6.  4a;2+9/  =  30 

7.  {y-lY  =  x^  8.  jt:2-y2=_-9 

The  curves  found  by  graphing  the  equations  just  given 
are  of  very  great  importance  in  the  work  of  the  world. 

The  curves  of  Exercises  1  and 
2  are  called  parabolas.  Such 
curves  are  often  used  for  the 
cables  and  arches  of  bridges.  A 
ball  thrown  into  the  air,  at  a 
slant,  rises  and  falls  in  a  para- 
bolic path.  The  curve  of  a 
paved  street  from  curb  to  curb 
■piG.  68  ^s   frequently  a  parabola.     The 

curves  of  Exercises  3  and  4  are 
circles;  Exercises  5  and  8  give  hyperbolas;  Exercise  6  gives 
an  ellipse  (Fig.  68),  which  is  frequently  used  for  bridge  arches. 
The  earth  moves  about  the  sun  in  an  elliptical  path. . 


i/ 

-^t       X 

^      ^  J  ^ 

^s^_^^ 

EXERCISE    II 


Solve,  drawing  the  graphs  of  the  first  three: 


1.  3;+2jc  =  5 

4.  45  =  ^?^ 
9  =  a+(w-l) 

7.  x-y=Vl 
^y=85 

10.  ic+2y  =  7 

M=5 

%    y 


2.  x^-^x+y^  =  (} 
x-y  =  () 

5:  a:2+y2=185 
x+y  =  \'7 

8.  w+2/>  =  5 
w24-2/>2=9 

11.  x+3>'-3  =  4>' 


3.  :r2-/=16 
Zx  =  by 

0.  x-\-y=l^ 
ic>'  =  36 

9.  2x-y  =  b 
x-\-3y  =  2xy , 

12.  ^-^^=4 


y- 


2 

x±Zy 
x+2' 


SETS  OP  EQUATIONS  IN  TWO  UNKNOWNS      267 


13. 

3x^-2xy=15 

2x-\-Sy=12 

14.  :r>;  =  72 
^  =  2 

15.     X2_y2  =  6 

:.-3'  =  3 

IG. 

3    fj(-fr+0 
2          2 

/  =  l+(/^-l)(- 

-tV) 

17.  4:^2+^-^4 
y+2x=l 

18.  y'^  =  2x+4 
3a:-2y  =  0 

10. 

ic^;  —  y  =  5 
:r-h:v  =  3 

20.  p+q=lOO 

1  +  1=1 

^^g     20 

21.  2s  =  n(a-^l) 
l=.a-\-{n-l)d 

22.  In  Ex.  21  determine  n  and  /  when  5  =  25,  a  =  l,  d  =  2. 

198.  Problems.  It  will  be  found  that  in  most  of  the 
following  problems  it  is  desirable  to  use  two  unknowns. 

1.  A  farmer  bought  160  acres  of  land  for  $16000.  If  part  of  it 
cost  $80  an  acre  and  part  $120  an  acre,  find  the  number  of  acres 
bought  at  each  price. 

2.  Two  boys  are  12  miles  apart  and  walk  toward  each  other,  one 
at  the  rate  of  2^  miles  an  hour  and  the  other  at  the  rate  of  3^ 
miles  an  hour.  How  far  will  the  first  boy  have  walked  when  they 
meet.    Give  both  algebraic  and  graphic  solutions. 

3.  A  dealer  made  the  following  offers:  Six  pounds  of  nuts  and 
4  pounds  of  candy  for  $4 .  50,  or  4  pounds  of  nuts  and  5  pounds  of 
candy  at  $4 .  40.  What  is  the  price  per  pound  of  the  candy  and  the 
nuts? 

4.  A  grocer  wishes  to  mix  2  brands  of  coffee  worth  25  cents  and 
40  cents  a  pound  respectively  How  many  pounds  of  each  must  he 
use  in  making  a  mixture  of  11  pounds  worth  35  cents  a  pound? 

5.  A  certain  brand  of  coffee  made  up  of  two  kinds  in  the  ratio 
1 : 2  formerly  sold  at  55  cents  a  pound.  The  first  kind  rose  20  per 
cent  in  price,  the  other  50  per  cent.  The  same  mixture  now  sells 
at  78  cents  a  pound.  What  is  the  present  price  of  each  kind  of 
coffee  used? 

6.  A  crew  that  can  row  10  miles  an  hour  downstream  finds  that 
it  takes  them  twice  as  long-  to  row  back  the  same  distance.  Find 
the  rate  of  the  current. 


268  •    BEGINNERS'  ALGEBRA 

7.  Suppose  you  were  camping  on  a  river.  How  could  you  find 
the  rate  of  the  current  of  the  river  and  your  own  rate  of  rowing? 

8.  A  man  has  $600  invested  in  Liberty  bonds,  a  part  at  4  per  cent 
and  the  rest  at  4  J  per  cent.  The  annual  income  from  the  bonds  is 
$26.    How  many  dollars  worth  of  each  kind  of  bonds  does  he  have? 

9.  Two  children  weighing  30  and  50  pounds  respectively  are 
riding  on  a  12-foot  teeter  board.  How  far  from  the  lighter  child 
should  the  support  be  placed  so  that  they  shall  just  balance? 

10.  Two  children  riding  on  a  teeter  16  feet  long  just  balance 
when  the  support  is  placed  10  feet  from  the  end  of  the  board. 
If  the  child  on  the  long  end  weighs  40  pounds,  what  does  the  other 
weigh? 

11.  A  teeter  is  supported  at  the  middle.  Two  boys,  one  weigh- 
ing 75  poimds  and  the  other  45  pounds,  wish  to  ride  on  it  so  that 
they  will  just  balance.  The  lighter  boy  got  on  the  end  of  the  board 
which  was  6  feet  from  the  support.  How  far  from  the  support 
should  the  heavier  boy  sit? 

12.  Two  boys  of  unknown  weights  found  that  they  balanced 
when  one  was  twice  as  far  from  the  fulcrupi  as  the  other.  Can  you 
find  out  how  much  they  weighed?  Can  you  find  their  relative 
weights?  If  the  lighter  boy  was  at  the  end  of  the  12-foot  teeter, 
where  was  the  heavier  boy?  * 

13.  The  heavier  of  the  boys  of  the  last  exercise  was  rather  bright. 
He  remembered  the  10-pound  sack  of  flour  he  was  taking  home. 
He  noticed  that  they  just  balanced  when  he  was  3  feet  from  the 
fulcrum  and  the  lighter  boy  was  at  the  end  of  the  teeter.  The 
lighter  boy  took  the  flour  in  his  lap  and  found  that  they  just  bal- 
anced when  he  moved  1-^  feet  nearer  the  fulcrum.  How  much 
did  each  boy  weigh? 

14.  Two  objects  of  unknown  weight  just  balanced  when  placed 
20  inches  and  12  inches  from  the  middle  of  a  rod  which  is 
balanced  at  the  middle.  If  the  objects  are  reversed  in  position 
it  is  necessary  to  add  4  pounds  to  the  lighter  object.  Find  the 
weights. 

15.  The  sum  of  two  numbers  is  27,  their  product  is  180.  What 
are  the  numbers? 


SETS  OF  EQUATIONS  IN  TWO  UNKNOWNS      269 

16.  Two  boys  are  carrying  a  weight  of  150  pounds  hanging  from 
a  9-foot  pole  between  them.  The  weight  is  6^  feet  from  one  boy 
and  2^  feet  from  the  other.     How  much  does  each  carry? 

The  principle  of  the  lever  can  be  applied  in  Problem  16  just  as 
well  as  in  teeter  problems.  The  place  where  the  weight  is  hung 
is  to  be  taken  as  the  fulcrum.  If  x  is  the  part  of  the  weight 
carried  by  one  boy,  how  much  will  the  other  boy  carry? 

17.  The  sum  of  the  squares  of  two  numbers  is  74  and  the  sum 
of  the  numbers  is  12.    What  are  the  numbers? 

18.  The  sum  of  two  nimibers  is  9  and  the  difference  between 
their  squares  is  135.    What  are  the  numbers? 

19.  The  area  of  a  rectangle  is  84,  while  its  perimeter  is  38.  What 
are  its  dimensions? 

20.  Divide  126  into  two  parts  that  shall  be  in  the  ratio  of  2  to  7. 

21.  There  are  two  numbers  whose  sum  is  20.  If  the  larger  is 
divided  by  the  smaller  the  quotient  is  5  and  the  remainder  is  2. 
Find  the  nimiber. 

22.  There  are  two  nimibers  whose  difference  is  6.  If  4  times  the 
larger  is  divided  by  5  times  the  smaller  the  quotient  is  1  and  the 
remainder  is  5 .     Find  the  numbers. 

23.  Find  three  consecutive  integers  such  that  the  product  of  the 
first  and  third  divided  by  the  second  gives  a  quotient  of  24  and  a 
remainder  of  24. 

24.  There  is  a  certain  ntunber  of  two  digits  such  that  if  the  nimi- 
ber is  divided  by  the  difference  between  the  digits  the  quotient  is 
21.    The  sum  of  the  digits  is  12.     Find  the  number. 

25.  If  a  is  added  to  both  the  numerator  and  the  denominator  of 
a  certain  fraction  the  result  is  equal  to  f.  If  b  is  subtracted  from 
both  the  numerator  and  the  denominator  the  result  is  equal  to  ^. 
Find  the  fraction. 

26.  What  will  the  answer  to  Exercise  25  become  if  (1)  a  =  5, 
5  =  3;  (2)  a=3,  6=5;  (3)  a=4,  6=1? 

27.  Find  three  consecutive  numbers  such  that  the  square  of  half 
of  the  middle  one  is  2^-  times  the  sum  of  the  other  two. 


270  BEGINNERS'  ALGEBRA 

28.  The  ntimerator  of  a  certain  fraction  is  one  less  than  the 
denominator.  If  4  is  subtracted  from  the  numerator  and  1  added  to 
the  denominator  the  value  of  the  fraction  is  -g-.    Find  the  fraction. 

29.  The  sum  of  the  nimierator  and  the  denominator  of  a  certain 
proper  fraction  is  13.  If  3  is  added  to  the  numerator  and  the 
denominator  is  multiplied  by  2  the  value  of  the  fraction  is  -J. 
Find  the  fraction. 

30.  If  2  is  added  to  the  nimierator  of  a  certain  fraction  and  2 
subtracted  from  the  denominator  the  value  of  the  fraction  is  yq- 
If  3  is  added  to  the  nimierator  and  4  is  added  to  the  denominator 
of  the  same  fraction  the  value  of  the  fraction  is  -J-.  Find  the 
fraction. 

31.  A  and  B  can  do  a  piece  of  work  in  6  days.  A  works  2  days 
and  B  3  days  and  in  that  time  they  do  f  of  the  work.  How  long 
will  it  take  each  to  do  the  work? 

Solution.    The  equalities  are:  - 

Part  of  work  A  can)  .f part  of  work  B  can"!  ^  (part  both  can  do 
do  in  1  day  i      1     do  in  1  day  i      1     in  1  day 

Part  A  does  in  2  days+part  B  does  in  3  days  =  f  of  the  work 
li  x=no.  of  days  it  would  take  A  to  do  the  work 

-  =  Part  of  work  A  could  do  in  1  day 

X 

2 

-= Part  of  work  A  could  do  in  2  days 

X 

The  equalities  above  expressed  in  algebraic  terms  are 

V=- 

X  y  6 

x'^y    5 

32.  A  and  B  undertook  to  address  a  certain  number  of  letters. 
They  knew  that  they  could  do  the  work  in  9  hours.  After  4  hours 
A  was  taken  sick.  It  took  B  10  hours  more  to  finish  the  job  work- 
ing alone.  Which  was  the  faster  worker?  How  long  would  it 
take  each  to  do  the  work  alone? 


SETS  OF  EQUATIONS  IN  TWO  UNKNOWNS      271 

33.  The  sum  of  two  fractions  having  the  numerators  2  and  5 
is  f.  If  the  denominators  are  exchanged,  the  sura  of  the  resulting 
fractions  is  2.    Find  the  fractions. 

34.  Water  is  being  taken  out  of  a  tank  by  a  pump  B  while  the 
tank  is  being  supplied  by  a  pipe  A.  When  both  pipe  and  pump 
are  running  the  tank  can  be  filled  in  12  hours.  The  tank  has  just 
been  cleaned.  The  pump  is  started  2  hours  after  the  pipe  was 
turned  on.  The  tank  was  filled  in  3|-  hours.  How  long  will  it  take 
the  pipe  to  fill  the  tank? 

35.  A  cistern  can  be  filled  by  two  pipes  running  together  in  2 
hours  and.  55  minutes;  the  larger  pipe  by  itself  will  fill  it  2  hours 
sooner  than  the  smaller  pipe  by  itself.  How  long  would  it  take 
each  pipe  to  fill  the  cistern? 


CHAPTER  XIII 
Radicals 

199.  Square  root  of  a  number.  The  square  root  of  a 
number  was  defined  as  one  of  the  two  equal  factors  of  a 
number.  This  was  used  in  chapter  x  as  the  basis  of  the 
method  of  finding  the  square  root  of  a  number. 

EXERCISES 

Find  the  following  square  roots  by  inspection: 

1.   V49  2.    VSI  3.  Vl6 

4.   -V25  5.   -Vo^  6.  Vl2i 

7.   V225  8.   V625  9.  -V36     • 

10.   -VIOO  11.   V?  12.  Vic* 

13.   -Va6  14.   -V9a2  I5.  Vl44 

200.  Square  root  of  monomials.  Finding  the  square  root 
of  a  number  is  the  reverse  of  finding  the  square  of  a  number : 

(3a^)2  =  32(a4)2  =  9a8 

To  square  a  number,  we  square  the  numerical  coefficient 
and  multiply  the  exponent  of  each  of  the  letters  by  2. 

To  find  the  square  root  of  a  product,  we  reverse  the  opera- 
tion; namely,  take  the  square  root  of  the  numerical  coeffi- 
cient and  divide  the  exponent  of  each  letter  by  2: 

VT6^«=  Vl6a^6^  =  4a63 

EXERCISES 

Find  the  following  square  roots: 

1.    Vio^  2.   -  V9x»  3.    VsiF 

4.    V25^2  5.   _  V64  6.    V16? 

7.    V25o«  8.   -4:^J9a^x*  9.    Vl6x'y* 

10.   V625w6  11.   -  ^lm<  12..  yJsm 

272 


RADICALS  273 

13.   -Vl96^  14.    V32i  15.    V576 

16.    V25^*  17.    ^Jix-^yy  18.    ^l9n*{w+l)^ 

19.   -  Vl21n«(a+6)2     20.    -  Vl69o2^  21.    V36(:x:-1)« 

201.  Bonds  of  numbers.  In  our  study  so  far  we  have 
come  upon  several  kinds  of  numbers,  among  which  are: 

Integers  such  as  2,  5,  27 

Fractions  such  as  tt,  «,  -^ 
o    7     0 

Irrational  square  roots  such  as  V2,   Vs,   Vl8 

Negative  numbers  such  as  —  3,— ^,  —  Vs 

o 

The  third  kind  belongs  to  a  much  larger  class  of  numbers, 
called  irrationals,  including  irrational  cube  roots,  fifth  roots, 
etc.,  such  as  \3,  V7,  and  other  more  general  irrationals 
such  as  7r  =  3.14159+. 

It  is  proved  in  later  coiu-ses  in  algebra  that  an  irrational 
number  cannot  be  expressed  exactly  as  a  fraction  or  as  a 
decimal.  We  therefore  use  approximate  values  when  using 
irrationals  in  arithmetical  calculations  as  was  done  in 
chapter  x. 

It  is  the  purpose  of  this  chapter  to  show  how  we  may 
combine  certain  .irrationals  without  using  their  approximate 
values. 

In  this  book  we  shall  consider  only  square  roots. 

202.  Definition.  A  root  indicated  by  the  use  of  the 
symbol  V    is  called  a  radical. 

V9^    Va2+2a6H-62 

VsT   ^la+x 
The  square  root  operation  can  be  completely  carried  out 
in  the  first  two  instances,  the  results  being  3  and  a +6.     In 
the  other  instances  the  operation  cannot  be  carried   to 
completion;  the  expressions  are  really  irrational. 


274  BEGINNERS'  ALGEBRA 

203.  Numerical  calculation.  Let  it  be  reqtiired  to  find 
the  numerical  value  of  the  following  combinations: 

(a)  v^+V2;  {b)  V3"-V2;  (c)  VFxVs;  {d)  V3"--V2 

The  results,  of  course,  are  wanted  in  approximate  decimal 
form  and  may  be  found  by  the  use  of  approximate  values 
for  V2^  Vs.     Carry  the  result  out  to  four  figures. 

V3"=1.732,    V2'=1.414 

(a)         V3'=  1.732  (6) 

+  V2"=  1.414 
3.146 

(c)  V3]=  1.732  (d) 

XV2=  1.414 
2.449 

Let  the  student  compute  in  a  similar  way  the  following: 

(e)  Vl8+V8;"(/)  VlS-Vs;  (g)  VlSxVs^  {h)  VIS  ^  Vs 

The  values  of  a/Ts  and  V  8  may  be  taken  from  the  table 
of  square  roots.  If  yotu*  calculations  have  been  made 
correctly,  the  results  are  for  (e)  7.071,  (/)  1.414,  (g)  11.999 
or  12.00,  (h)  1.5000  or  1.500.  All  four  of  these  can  be 
calculated  by  simpler  methods,  which  will  be  developed  in 
the  following  articles. 

204.  Addition.  There  is  no  other  way  of  calculating 
V2"  +  Vs* than  that  used  in  the  last  article.  But  VlS  +  Vs 
presents  possibilities.  We  can  find  the  square  root  of  a 
product  by  finding  the  square  root  of  each  factor  and 
multiplying  the  square  roots : 

Vl8=V^^=3V2 
So  also  Vs"  =  VT^  =2  V2 

hence  VlS  +  Vs"      *3V2'-|-2V2 

=  5V2 
=  5X1.414  =  7.070 


RADICALS  275 

Notice  that  3  and  2  may  be  considered  as  the  coefficient 
of  V^. 

This  is  not  quite  the  same  as  the  result  found  in  the 
other  way,  but  the  error  is  no  more  than  might  be  expected 
when  one  is  working  with  approximate  numbers. 

There  is,  moreover,  another  aspect  of  the  matter  that  is 
often  of  as  much  importance  as  the  finding  of  the  numerical 
value.  The  sum  of  the  two  radicals  has  been  expressed  in 
a  more  concise  radical  form : 

5  V2  rather  than  VTs  +  Vs 

205.  Simple  radicals.  The  forms  of  radical  expressions 
can  be  altered  without  change  in  their  values.  5V2  is  an 
example  of  what  is  called  a  simple  radical.  Vl8  is  not  a 
simple  radical.  There  is  concealed  in  the  18  a  square 
factor,  Vl8  =  V9^  =  3  V2  .  Find  Vl8  and  3  V2  and  compare 
answers.  The  square  root  of  an  integer  or  an  integral 
expression  that  has  no  square  factor  is  called  a  simple 
radical.  Two  radicals  that  have  the  same  number  under 
the  radical  sign  are  called  like  radicals.  The  possibility 
of  adding  two  radicals  into  one  term  depends  upon  whether 
or  not  they  are  like  radicals. 

EXERCISE    I 

Reduce  to  simple  radicals.  Verify  1—6  as  you  verified  Vl8= 
3V2 .    In  22  to  28  assign  values  to  the  letters  and  work  out. 


1.  V50 

8.  V72 

15.  VI8O 

22.  V5^=^ 

2.  V12 

9.  Vl28 

16.  V24 

23.  Vi^ 

3.  V27 

10.  Vl62 

17.  V54 

24.  Va2Z> 

4.  Vs 

11.  V2OO 

^       18.  V96 

25.  V49:x; 

5.  V75 

12.  Vis 

19.  Vl50 

26.  ^J81x^y 

6.  V45 

13.  Vl47 

20.  V28 

27.  ^l3a'^x 

7.  3V98 

14.  V125 

21.  V90 

2S.  5V32a» 

276 


BEGINNERS'  ALGEBRA 


To  work  the  following  sums  reduce  to  simple  radicals,  and  find 
the  numerical  value  of  the  results  by  use  of  the  tables.  Check  each 
exercise  by  use  of  the  tables  without  reducing  to  simple  radicals. 


Illustration: 

Check: 

V5O-VI8  +  V3 

=  5V2-3V2+V3 
=  2V2+V3 
=  2(1. 414) +  1.732 
=  2.828+1.732 

V50  =  7.071 

+  V3  =  1.732 

8.803 

-Vl8  =  4.243 

=4.560 

4.560 

EXERCISE   II 

1.  2+V3 

2. 

2V3- 

-5V3+7V3 

3.  V12+V27 

4. 

Vso- 

-vr5 

5.  VI2+V75- 

-V27 

6. 

VTs- 

-V2-V8 

7.  2V3-Vl2+*V48 

8. 

2V2- 

-3V3+5V3-6V2 

9.  V18-V2+V32-5V5 

10. 

V98- 

-V50+V32 

Carry  out  the 
following: 

indicated  operation  whenever  possible  in  the 

11.  2^/a-S^la- 

-7Va 

12. 

V3^- 

-Vl2a  +  V48a 

13.  ylab^+^Jab^ 

14. 

Va6'  +  Va36 

15.  V9x-V4^  +  V25i 

16. 

2V3^ 

-^^Jl2x-{-^|7Ex 

17.  aylb-cylb 

18. 

a-V 

a^b+^'^b 

206.  Addition  involving  fractions.    Add: 
>f +V24 

You  can,  of  course,  find  -^?  as  in  chapter  x,  but  here 


we 


wish  to  see  if  we  can  combine  these  two  radicals  into  one. 
There  is  a  better  way  of  proceeding.    This  is  accomplished 


RADICALS  277 

by  a  very  simple  device.     Alter  the  fraction  |  so  that  the 
denominator  is  a  square  by  multiplying  both  terms  by  3. 

Thus, 


^3"  >'3X3     \9       3   ~3^^ 
Hence  ^|'+V24  =  iV6+2V6 


(|+2)V6 


xV6 


The  same  method  will  work  with  any  radical  that  has  a 
fraction  under  the  radical  sign: 


VrV^^-;^ 


Supply  values  for  a  and  b  and  work  out  the  result  by  this 
method. 

To  be  a  simple  radical,  a  radical  must  have  neither  a 
fraction  nor  a  square  factor  under  the  radical  sign. 

EXERCISE    I 

Reduce  to  simple  radicals  and  find  numerical  values  where  pos- 
sible. 


w-r 

-  ivr 

,3.  VI 

■'■Vi 

wr 

'■VI 

H.  Vl- 

^vt 

'Vr 

'■VI 

"■4 

2,.  ^=1 

.'■'4 

■«VA 

■«Vl 

^■V? 

5.7^1 

'■■VI 

"■•VI 

a  Vf 

e-Vl 

-Vi 

-V* 

^■iVf 

278  BEGINNERS'  ALGEBRA 

EXERCISE   II 

Complete  the  following  additions  and  subtractions: 


1.  V4-Vi 

2.  Vf +V54+Vf 

3.  Vi-Vf +  V50 

4.  VJ+Vi+VS+V? 

5.  Vi-V2+Vf 

6.  Vi+ViV-Vf 

7.  VI- Vf 

8.  Vf  +  VI 

207.  Multiplication. 

We  have  already  used  the  general 

principle 

Va6  =  VaVfe 
As  long  as  a  and  b  are  positive  numbers,  the  reverse  of 
this  is  equally  true: 

that  is,  V2V3  =  V6 

Test  this  result  by  calculation,  using  the  tables. 
So  VT8V8=Vli4  =  12 

Work  out  other  illustrations  of  the  use  of  this  principle 
by  suppl3ring  values  for  a  and  b. 

Which  method  is  to  be  preferred  for  calculation,  the 
method  used  here  or  the  method  used  in  Art.  203? 

Find  product  of  2V3  X  7 Vs 

State  a  rule  for  multiplying  radicals. 

EXERCISES 

Find  the  following  products.  In  Ex.  13,  14,  15,  16,  and  19 
assign  values  for  the  letters  and  work  out. 

2.  V5  V7 
4.  VS  V2 
6.  V3  V7 
8.  Vf  Vl5 

10.  V24  Ve 


1. 

V2  Vs 

3. 

Vi  V3 

5. 

V3  V27 

7. 

V6  Vl5  VlO 

9. 

Vf  Vj 

RADICALS  279 

11.  V|V45  12.  2V7-3V2 

13.  VaV2a  14.  ^J2a  ^l^x 

15.  V3V2a  16.  ^l^a^|2a 

17.  ^J2a  yJSa  18.  yf2ab  V2^ 

19.  VoT^  Va^  20.  V^H^  y/x+y 

21.  (2+V3)(2_^V3)  22.  (2  +  V3)(2  +  Vs") 

23.  3V2^-  -iVS^  24..  (V2 +V3)(V2'+V3") 

25.  (V3+V5)(V3 -Vs")  26.  (Vs  +  V2)(- V3  -  V2) 

27.  (V6-Va)(V6-V2)  28.  (Vl5-V5)(Vi5+V3) 

208.  Division.  We  may  obtain  the  numerical  value  of 
V3  -^  V2  as  in  Art.  203  by  dividing  1 .  732  by  1 .  414.  It  will 
be  remembered,  however,  that  in  Art.  173  we  used  the 
principle 

b      yfb 

This  principle  is  equally  true  if  read  in  the  reverse  direc- 
tion: 


Va  _  ^ja 


Using  this  idea  here,  we  have 

V3      .^/3     ^/6     1  r- 


V2        ^2       ^^4     2^ 

which  is  very 

easily  computed. 

So  also 

Vl8     ^/18     ./9     3 
V8~^8~^4"2 

or  again 

^-V.=3 

Give  other  illustrations  of  the  use  of  this  principle  by 
supplying  values  for  a  and  b. 


280  BEGINNERS'  ALGEBRA 

This  method  of  division  is  very  simple  and  direct  in  many 
cases.  Another  method  is  often  used  that  in  some  cases  is 
even  more  direct.     The  division  Vs  -^  V2  is  regarded  as  a 

V3 
fraction  — =-  • 

V2 

The  form  of  this  fraction  is  now  changed  to  some  form 
that  is  particularly  desired.  For  some  purposes  it  is  desir- 
able that  there  shall  be  no  radical  in  the  denominator.  To 
secure  this  result  multiply  both  terms  by  a  number  that 
will  bring  the  desired  form. 

Vs  .    .  I- 

In  the  case  —j=  it  is  easily  seen  that  V2  is  the  proper 

multiplier : 

V3xV2     yfQ  .     1   /- 


V2XV2       2         2 
For  some  purposes  the  numerator  should  be  free  of  radicals. 
What  multiplier  should  be  used  to  secure  that  result? 

We    call    this    process    of    freeing    a    term    of    radicals 
rationalizing  that  term. 

What  multipliers   will  rationalize  the  denominators  of 
3        3         V2      V3      VT? 


the  fractions 


V3    2V3     V5    Vs    V50 


EXERCISES 

In  each  exercise  use  the  method  you  think  best  suited  to  that 
particular  exercise: 

1.  VsOa^VlOrt  2.  VlG^Vs 

3.  V5      -^V2  4.  ^|7a^^J2b 

5.  3V2    -2V3  6.  V3^-^V2^ 

7.  V7     -j-Vs  8.  Ve   -^Vl2 

9.  2V5    -^V6  10.  (2V6-10V2'-h6Vl8)  -^V3 

11.  VI      ■^V|  12.  (V2"-V3)-^V6 


RADICALS  281 

209.  Division  by  a  binomial.  The  same  general  pro- 
cedure is  to  be  followed  when  the  divisor  is  a  binomial. 
The  division  is  to  be  expressed  in  the  fractional  form  and 
the  fraction  reduced  to  the  form  desired.  If  it  is  desired 
to  free  the  denominator  of  radicals,  a  satisfactory  multiplier 
must  be  sought.     A  new  problem  is  presented  here. 

Take,  for  instance,  —;= 1=^ 

V3+V2 

If  the  multiplier  Vs  +  V2  is  used,  we  have 

3(V3+V2) 
3+2V6+2 

The  desired  end  has  not  been  accomplished.  A  familiar 
identity  furnishes  a  clue  to  what  we  want: 

(a+6)(a-6)=a2-62 

Can  you  follow  the  clue  and  determine  the  multipUer 
needed  to  free  the  denominator  of  radicals  and  reduce  the 
fraction  to  the  form  required?  Formulate  a  rule  for  reduc- 
ing such  fractions  to  equivalent  fractions  with  denominators 
free  of  radicals. 

3 

Compute  the  value  of  the  fraction  -t= ip^  by  the  direct 

use  of  the  table  and  also  by  rationalizing  the  denominator. 
Get  a  result  showing  three  significant  figures  and  draw  your 
own  conclusion  as  to  the  two  methods. 

The  related  forms  Va  +  V6  and  Va  —  V6  are  called 
conjugates  of  each  other.     Their  product  is  rational. 

(Va+V6")(Va-V6)  =  a-6 

(V5+V3)(V5-V3)  =  5-3  =  2 

What  is  the  conjugate  of  V2  -  Vs   ?  of  V? -f  Vc   ? 

Give  other  illustrations  by  supplying  values  for  a  and  b. 
19 


282  BEGINNERS'  ALGEBRA 


EXERCISES 

Reduce  to  equivalent  fractions  with  rational  denominators. 
Where  possible  find  the  numerical  values.  Supply  values  for  letters 
in  Ex.  6,  7,  and  8. 

1.^^  2.-_l 


2+V3  *  V3-V2 

3.      i_^  4.    2V2 


7. 


V3-2  2+V2 

g    V3+V2^  g    ^lx-^|a 

^|S  -V2  '  V^+Va 

^'  8     ^~y 

9    ^^8+2|  V8-2  iQ    I-W2; 

.  '  VS  -2     V8+2  '  l+-i-V2 

11.  -L  +   1  12.  :^ 

2+V2      2-V2  V2-I 

210.  Practical  computation.  After  all  the  foregoing  dis- 
cussion what  do  you  think  would  be  the  most  feasible  and 
common-sense  way  of  computing  the  value  of 


v^«v 


257 
39 


1— ^^ 
,  -,  KiZ.O 

is  desired. 

Find  the  value  in  the  most  satisfactorj^  way  you  know. 

211.  Irrational  equations.  In  the  solution  of  problems  it 
often  happens  that  an  equation  appears  with  the  unknown 
imder  a  radical  sign.  Such  equations  are  called  irrational 
equations.  A  few  simple  equations  of  this  kind  wall  be 
considered  here.     One  of  the  simplest  is 

V^=3  (1) 


RADICALS  283 

Inspection  leads  at  once  to  the  answer 

x  =  9  (2) 

In  that  very  brief  inspection  you  really  squared  the  3  and 
likewise  the  Va;. 

This  brings  to  light  another  operation  that  can  be  appHed 
to  equations;  namely,  both  sides  of  the  equation  may  be 
squared  and  the  relationship  of  the  sides  to  each  other 
remain  undisturbed.     Apply  to  the  equation 

Squaring  both  sides,  we  have 

x-^-S^Jx-^-\-lQ==0 
an  equation  that  still  contains  a  radical  and  is  even  more 
complicated. 

The  object  of  squaring  was  to  remove  the  radical.  For 
this  purpose  the  radical  should  stand  alone.  Alter  the 
equation  so  that  the  radical  will  be  alone : 

ylx-S  =  4: 
and  then  square,  x  —  Z  =  lQ 

Check:  ^lW^  4 

Vl6  4 
4  4 

Apply  to  re— V^+2  =  0  and  check 

Do  both  roots  obtained  from  the  derived  integral  equation 
satisfy  the  original  irrational  equation?  How  many  roots 
does  this  particular  irrational  equation  have? 

Occasionally  you  find  answers  that  will  not  check.  These 
are  not  roots  of  the  equation.  This  matter  will  be  dis- 
cussed in  a  later  course. 

Formulate  a  rule  for  solving  irrational  equations. 

Apply  rule  to  solving 

x-yJx+2  =  2 


284  BEGINNERS'  ALGEBRA 

EXERCISES 
Solve  the  following  equations: 

1.  v^-5=o  2.  vr^=5 

3.  ^lx^-9-^  =  0  4.  w+Vn+6=14 

5.  2  =  3w+V5w2+ll  6.  n  -V»+6=14 

—  22 

^.     Find  value  for  m  when  <=  1,  7r  =  -y ,  and  g=32.2. 


■=V'4 


8.  C  =  ^  / 1  _i- i.     Solve  for  /.     Find  I  when.  C = 2. 

9.  a;+Vx^  =  2         10.  V:c'+7  =  0  11.  ^x^=S 

12.  ^+V^r^  =  15        13.  V^T5-a;  =  3  14.  V^^+10=x 

212.  Checking   solutions    of    quadratic    equations.    The 

roots  of  the  equation 

are  2+V2,     2-V2 

In  chapter  x  these  were  checked  by  the  use  of  approxi- 
mate values,  and  of  course  the  checking  could  not  be  exact. 
If,  however,  the  radical  forms  are  substituted,  the  checking 
is  exact : 

(2+V2')2-4(2+V2  )+2  0 
4+4r^f^+2-8-^^/?-f2  0   * 
0  0 

EXERCISES 


Solve  and  check: 

1.  4:r2-l  =  4:r 

2.  5««+ll  =  ir):v 

3.  .+3«L1 

4.  4:K2_i2a:+9=0 

THE  APPENDIXES 


THE  APPENDIXES 

APPENDIX  A 
FACTORING 

213.  The  sum  of  two  cubes;  the  difference  between  two 
cubes.  The  following  identities  may  be  verified  by  doing 
the  indicated  multiplication: 

(a^-ab-\-b^)(a-\-b)=a^+b'  (1) 

{a^-\-ab+b^){a-b)  =a^-b^  (2) 

Read  from  left  to  right,  these  give  special  types  of  multi- 
pHcation.  Read  from  right  to  left,  they  reveal  types  of 
expressions  that  can  be  easily  factored. 

When  put  in  words,  (1)  as  a  guide  to  factoring  reads: 
The  sum  of  the  cubes  of  two  ntmibers  equals  the  sum  of 
the  numbers  times  the  square  of  the  first  number  minus 
the  product  of  the  two  ntunbers  plus  the  square  of  the 
second  number. 
^  State  (2)  in  words  as  a  guide  to  factoring. 

You  should  notice  the  ways  in  which  the  two  identities 
are  alike  and  also  the  ways  in  which  they  differ. 

EXERCISES 
Factor: 

1.  a^-¥  2.  c3+8 

3.  a;3-125  4.  W3-64 

5.  r3+9  6.  r^+27 

7.  «3+27^3  8.  x^y^-t^ 

9.  l-125/f3  10.  53+1 

11.  /3+64  12.  27n^-Sp' 

13.  125+27^3  14.  64a3+729 

15.  25x^-200y^  16.  nH-\-nt* 

17.  TTi^a-Trr'  18.  4(^-S2bH^ 

19.  l+16c»  20.  16+54n« 

287 


288  BEGINNERS'  ALGEBRA 


21.  »«-27» 

22. 

8s'-^s 

23    ^  +  *-' 
27  ^  8 

24. 

4-  '• 

25.|a3  +  l6> 

26. 

f- 

Expand: 

27.     (5-/)(52  +  5/  +  ^2) 

28. 

(y2-t-5y+25)(y-5) 

29.  (2-;j)(4+2w+w2) 

30. 

a+4)(/2-4/+16) 

31.  (/j2-2w+4)(w+2) 

32. 

(x2_:c+l)(x+l) 

Fill  in  blanks: 

33.  (x2+?+49)(x-?)  =  x3-?3  34.  {x^+?x-\-9){?)  =  x^-?' 

35.  (y+?)(/-?+36)  =  3;3+?3  36.  (:x;2_^^_^?)(^_|.p)  =  ^34_i 

37.  {?-\-k)i25-?+k^)  =  ?'+k^  38.  (724-7-9+92)(9-7)  =  ? 

39.  {x^-Sxy-\-9y^){x+3y)  =  ?  40.  (2x-33')(4jc2+6:«y+9y2)  =  ? 

214.  Factoring  by  grouping  terms.  Sometimes  an  expres- 
sion that  does  not  at  first  sight  seem  to  conform  to  any  of 
the  factorable  types  known  to  you  can  be  so  altered  by  a 
judicious  grouping  of  terms  as  to  reveal  a  known  type. 
There  are  two  special  cases  of  importance  which  will  be  con- 
sidered in  the  next  two  articles. 

215.  A  common  binomial  factor  revealed.  Consider  the 
expression 

ax-\-ay-\-bx-\-by 
Grouping  the  first  two  terms  and  the  last,  we  have 

{ax + ay)  +  {bx + by) 
Factoring  each  group  separately,  we  have 
a{x-\-y)-i-b(x+y) 

In  this  form  is  revealed  a  binomial  which  is  a  common 
factor  of  the  two  terms  of  the  expression. 

The  expression  is  seen  to  be  of  the  common  factor  type  and 
may  be  put  in  the  form 

{x+y){a+b) 


APPENDIX  A 


289 


So  also 


=  f{t-2)-3(t-2) 
=  (f_2)(^2_3) 


EXERCISES 


Factor: 
1.  aix-\-y)  -b{x+Y) 
3.  xix-S)-{-5ix-S) 
5.  Sa{y-z)+Ab(y-z) 
7.  a^-{-ab+ac+bc 
9.  x^-{-2ax-{-3bx-\-Qab 
11.  x^-2x^-\-3x-6 
13.  h^-\-h^k-{-hk^+k^ 
15.  acn'^—bcn-\-adn—bd 
17.  a;3-12-4:^4-3.T2 
19.  xy— ay— 6;x;+a6 
21.  3;3-4>;-3y2+12 
23.  15ic3-12:r2+35:c-28 
25.  x^-x'^-^x-l 
27.  a6:*;+3x-10a6-30 


2.  ^h-k)-a{h-k) 
4.  :r(a-6)-4(a-&) 
6.  2x(:x;+2)  -3(.i:+2) 
8.  3:r3+2x2+3:t:+2 
10.  a:c2-6x24-a3;2-6y2 

12.  bx^-x-'-bx^-l 
14.  6/2+3/5 -2a^ -05 
16.  10a:3+2:x:-25:c2-5 
18.  x^-a'^b-a'^x'^-^x'^b 
20.  3x3-4jc2-6:x:+8 
22.  18a3+12a2-15a-10 
24.  9x3-x2-9:r+l 
26.  a3-7a2-4a+28 
28.  2a3-3a2-4a+6 


Con- 


216.  The  difference  between  two  squares  revealed. 

sider  the  expression 

a2+2a6+62-c2 

The  first  three  terms  are  at  once  recognized  as  a  trinomial 
square:  {a}-{-2ah-\-h'')  -c" 

The  expression  may  thus  be  put  in  the  form 

which  is  the  difference  between  two  squares,  and  we  have 

[(a+6)+c][(a+6)-c] 
or  (a+6+(;)(a+6— c) 

So  also 

i;c2- 10^+25 -:v2  =  (^2_  10^+25) ->;2 
=  (:;c-5)2-/ 


290 


BEGINNERS'  ALGEBRA 


So  also 

c2-a2+2a6-62 

The  last  three  terms  have  the  form  of  a  trinomial  square 
with  the  single  exception  that  the  signs  of  the  square  terms 
are  minus  instead  of  plus.  We  can  rectify  this  by  putting 
them  in  parentheses,  thus: 

c2-a2+2a6-62  =  t2_(^2_2a6+62) 

=  c^-{a-by 

=  [c-{-{a-b)][c-(a-b)] 

=  (c+a—b)(c—a+b) 


EXERCISES 


Factor: 

1.  {a -by -4: 

3.  y^-(x+2y 

5.  (2x-3)2-&2 

7.  (a -6)2 -25^2 

9.  4:9a* -{a -by 
11.  {4:X-2yy-lQx^y^ 
13.  9ix+2yy-25{x+ay 
15.  x^-2xfl-y'' 
17.  l+2ab-a^-b^ 
19.  4:-n^-p^-2pn 
21.  4:-4n-\-n\-n* 
23.  x^-\-Gxy-\-9y^-25 
25.  y^-4x^-25-20x 
27.  9:c2-30:r+25 -16:^:4 
29.  a^-'C^-d^-\-4+2cd-4:a 


2.  9-{x-yy 

4.  a2-(4x-5)2 

6.  (a;-y)2-(a+6)« 

8.  Wa'^b^-ix^-ay 
10.  (3a+8)2-36a* 
12.  4(a-6)2-(x-y)2 
14.  {x-\-3y-9y^ 
16.  a;2-a2+2a6-62 
18.  x^-\-y^-z^-2xy 
20.  a^-]-b^-c^-d^-\-2ab+2cd 
22.  A4+4/|3+4/j2-9 
24.  :x;2-14:*;+49-4y« 
26.  y2_j_4:^2_4^3,_25 
28.  a^-\-b^  -c^  -d^  -2cd-{-2ab 
30.  :x;2H-9y2-a2-62-6xy+2a/> 


217.  Factoring  after  expanding.  Sometimes  an  expression 
may  be  reduced  to  a  factorable  form  by  expanding  it  first. 
Consider 

(a+6)2-4a6  =  a2+2a6+62-4a6 
=  a2-2a6+62 
=  {a-by 


APPENDIX  B 


291 


The  example  just  given  illustrates  the  fact  that  it  is  not 
always  possible  to  get  the  original  expression  by  multi- 
plying together  its  factors,  although  the  product  so  obtained 
is  always  equal  to  the  original  expression,  for 

(a-6)2  =  a2_2a6+62 
which  is  not  the  original  expression. 


EXERCISES 


Factor: 

1.  4:ad+{a-dy 

3.  a(a-24)+63 

5.  {a-{-by-4ab 

7.  5x^-\-l-\-Sx{5x-S) 

9.  4a(3o-8)+5 
11.  4:x{x—y)—5y{4:X—7y) 
13.  6x+(x-Sy-18 
15.  2(x-l)(x-{-l)+3x 
17.  {x-8)ix-2)-\-2x^-13 
19.  x^-l-ix-1) 
21.  4a;2+(a;2-2)2-5 
23.  (x^-2){x^-\-2)-3x^ 


2.  {x-\-iy-5x-^0 

4.  (ic+7)2-28:«; 

6.  2oxix-y)+5{5x-^y) 

8.  3:»:(3jc+10)+25 
10.  {n-7){n+2)-\-n^+ll 
12.  kik-l)-cic+l) 
14.  {x-3)^-d(x-2)-h5 
16.  (:,+2)(:«-l)-(3^+l) 
18.  Sx^-{x-2){x-\-2)+5x 
20.  (2x+l)^-3x{x-{-2) 
22.  (a;2+6)2-25jc2 
24.  2x^''xix-\-l)-Q 


APPENDIX  B 
218.  Square  roots  of  polynomials.  It  is  possible  to  find 
the  square  root  of  many  algebraic  expressions,  that  is,  to 
express  them  as  the  product  of  two  equal  factors.  Only 
the  simplest  cases  arise  in  common  practice.  The  square 
roots  of  these  can  easily  be  found  by  inspection.  These 
have  already  been  considered  in  Art.  118.  One  has  but 
to  recall  the  simple  identity 

(a+6)2  =  a2+2a6+62 


'^4x^-12xy-\-9y^=  ^l{2x-^yy 
-2.r-3y 


292  BEGINNERS'  ALGEBRA 

EXERCISES 

Find  by  inspection  the  following  square  roots: 
1.  x^-^x+4:  2.  a^-Qab-\-9b^ 

3.  y-lOy+25  4.  x^-{-36-{-12x 

5.  ^9+n^+Un  6.  4ic2- 12^+9 

7.  25+36x^-'mx  8.  x^+8x^-{-lQ 

219.  Square  roots  of  polynomials  by  the  division  method. 

The  method  of  inspection  is  not  easily  .adapted  to  finding 
the  square  root  of  longer  polynomials  than  those  considered 
in  the  last  article. 

A  method  has  been  devised  for  finding  the  square  root 
in  such  cases,  but  it  is  seldom  required  in  mathematical 
work.  It  is  the  same  method  that  is  commonly  used  in 
arithmetic  for  finding  the  square  root  of  a  ntunber. 

The  method  is  founded  on  the  relation 
a2+2a6+62  =  (a+6)2 

or  ^la^-\-2ab+b^  =  a+b 

It  is  to  be  observed  that  the  first  term  of  the  root  is  the 
square  root  of  the  first  term  of  the  number;  that  is,  a  is 
the  square  root  of  a^.  If  the  square  of  a  is  subtracted  from 
the  number,  there  remains 

2ab+b^ 

The  first  term  of  this  remainder  is  the  product  of  twice 
the  first  term  of  the  root  and  the  second  term  of  the  root. 
This  suggests  that  we  may  find  the  second  term  of  the  root 
by  dividing  the  remainder  by  twice  the  first  term  of  the 
root  as  a  trial  divisor. 

Now  notice  that  the  remainder  may  be  factored: 
2ab+b^={2a+b)b 

This  shows  that  the  trial  divisor  2a  must  be  completed 
by  the  addition  of  6,  which  makes  the  complete  divisor 

2a-\-b 

If  this  is  then  multiplied  by  b,  the  second  term  of  the 


APPENDIX   B 


293 


root,  and  subtracted  from  the  remainder,  nothing  remains, 
and  the  root  a +6  is  obtained. 

An  illustration  or  two  will  show  the  process  to  the  best 
advantage : 

Illustration  1: 


7x-5 

49;x;2-70:«+25 

o2 

49^8 

Trial  divisor 

2fl 

=       Ux 

-70ic+25 

Complete  divisor  2a 

oot 

=       14:r-5 

-70:r-f25 

Tjc  —  5  is  the  square  i 

Illustration  2: 

Jc2+2a;+3 

n2           — 

^+4jc3+10x2-|-12:x;+9 

^4 

2a       = 

2^2            ^x^-\-lOx^ 

2a+6= 

2^24-2^    4jc'-l-4;t:2 

2a       = 

2:^2+4x            6:c'-4-12:»;+9 

2a+6  = 

2x2+4a:+3 

6.r2+12jc+9 

II 


The  fundamental  process  is  repeated  until  the  expression 
whose  root  is  to  be  found  is  used  up. 

The  following  figures  illustrate  the  reasons  for  the  method 
from  a  graphic  point  of 
view. 

I  represents  the  ex- 
pression. 

II  represents  the  first 
remainder. 

III.  The  line  2a  repre- 
sents the  trial  divisor; 
the  base  of  the  rectangle 
represents  the  complete 
divisor. 


oS 

6' 

ab 

b' 

a' 

ab 

ab 

III 


ab       ab     b 


2a  — 

Fig.  69 


294 


BEGINNERS'  ALGEBRA 


EXERCISES 

Find  the  square  roots  of  the  following  polynomials: 

1.  x*-^2x^-\-3x''+2x^l  2.  4x4+12ic'4-13:c2+6x+l 

3.  »8-2n6+3w«-4w3+3w2-2«+l      4.  ^y*-12y^-7y^+2^y-{-lQ 

220.  Square  root  of  arithmetical  numbers.  The  square 
root  of  a  number  like  3721  can  be  found  by  the  division 
method  used  in  the  last  article.  A  simple  modification  is 
needed  in  order  to  determine  what  corresponds  to  the  a^, 
the  first  term  of  the  number.     Is  it  3  or  37  in  this  case? 

The  square  of  a  number  of  1  digit  has  1  or  2  digits. 

The  square  of  a  number  of  2  digits  has  3  or  4  digits. 

The  square  of  a  number  of  3  digits  has  5  or  6  digits. 

The  reverse  statements  are  true: 

A  number  of  1  or  2  digits  has  1  digit  in  its  square  root. 

A  number  of  3  or  4  digits  has  2  digits  in  its  square  root. 

A  number  of  5  or  6  digits  has  3  digits  in  its  square  root. 

Hence  we  mark  off  the  numbers  in  pairs  of  digits  begin- 
ning at  the  decimal  point  and  marking  each  way. 

60+1 


a'       = 

2a      = 

b  = 

2a+b  = 


37  21 
36  00 


120 

1 

121 


121 


121 


The  square  root  is  61. 


a2  = 

2a  = 

b  = 

2a+b  = 

2a  = 

b  = 

2a+b  = 


200+30+2 
5  38  24 
4  00  00 


400 
30 

"430 


138  24 


1  29  00 


460 
2 

462 


9  24 


9  24 


APPENDIX 

B 

or,  omitting  the  useless  zeros, 
2    3    2 

3    5  .7 

5  38  24 

12  74  .49 

40 

138 

60 

3  74 

3 
43 

129 

5 
65 

3  25 

46 

)0 

9  24 

700 

49  49 

4( 

2 

9  24 

7 
707 

49  49 

295 


If  the  figures  of  the  root  are  placed  above  the  proper 
group,  there  will  be  no  difficulty  in  determining  where  the 
decimal  point  goes. 

EXERCISES 

Find  the  square  roots  of  the  following  numbers: 
1.  1156        2.  17424        3.  59049  4.  .0841 

5.  5329        6.  4489  7.  151.29        8.  4.9284 

221.  Approximate  roots.  The  same  method  is  used  in 
finding  approximate  values  for  irrational  roots.  The  only 
difference  is  that  the  process  does  not  stop,  but  may  be 
carried  on  as  far  as  one  chooses,  zeros  being  annexed  to 
the  remainders  as  the  work  continues. 

Find  V3": 

1.732 


20 

_7 

27 

340 

3 

343       

3460  I  71  00  etc. 


3 

.0000 

1 
200 

189 

11  00 

10  29 

296  BEGINNERS'  ALGEBRA 

EXERCISES 

Find  the  following  square  roots  to  four  figures: 
1.  V2~  2.  V5"  3.  vis 

4.  V32  5.  Vr  6.  V115 

APPENDIX  C 

EXPONENTS 

222.  Positive  integral  exponents.  We  have  been  using  a 
positive  integer  as  an  exponent  to  indicate  the  number  of 
times  a  factor  is  to  be  used  in  a  product. 

a^  means  aaaaa 
We  have  worked  with  these  exponents  according  to  certain 
fixed  rules  or  laws. 

I.  In  multipl5dng, 

In  multiplying  we  add  the  exponents  of  the  same 
letter. 
II.  In  finding  a  power, 

In  finding  a  power  we  multiply  the  exponent  of  the 
letter  b}''  the  power  to  be  found. 

III.  In  dividing, 

a^  -^  a^  =  a^-^  =  a^ 

In  dividing  we  subtract  the  exponent  of  the  divisor 
from  the  exponent  of  the  same  letter  in  the  dividend. 

IV.  In  finding  a  root, 

In  finding  a  root  we  divide  the  exponent  of  the  num- 
ber by  the  number  indicating  the  root. 

223.  Some  peculiar  expressions.  In  applying  rules  III 
and  IV  we  come  upon  several  very  peculiar  expressions. 


APPENDIX  C  297 

Va^  =  a^ 

224.  Extension  of  exponent  idea.  If  we  are  to  be  per- 
mitted to  use  Rules  III  and  IV  as  in  the  last  article,  we 
must    find   some  meaning  for    such   expressions  as  a^yOr^^ 

and  a\ 

225.  Meaning  of  zero  exponent.  The  meaning  of  the 
zero  exponent  has  been  shown  in  Art.  141. 

By  ordinary  division       i"  ^ 
Therefore  a9  =  l 


EXERCISES 

Find  value  of:    3«,    a^,    2a<>,    2xy'>,    ah\    a^¥-\-2ab-{-a^b^,    27o, 
1921°,  3X100». 
226.  Meaning  of  negative  exponent. 

But  by  reducing  to  lowest  terms 

Therefore  a-2=— 

A  negative  exponent  means  that  the  number  over  which 
it  is  placed  is  to  be  used  as  a  divisor  rather  than  as  a  factor. 

EXERCISES 

1.  Find    the   value   of:    2\  Z\  a-\  a^  3a  ^  2a-\  2ah-*,  lO'S 
2x10-2,  2.5X10',  3X10«;  2-1+3-S  2x3°X5-2. 

2,  Express  with  negative  exponents:    — ,  — ,  — ,   — ,    .00003, 
.00000062. 

20 


298 


BEGINNERS*  ALGEBRA 


3.  Express  without  negative  exponents:  a'^,  a-',  ab'^,  2ab^, 
Sab-^x-^. 

227.  Meaning  of  a  fractional  exponent. 

Let  us  see  what  happens  if  we  apply  the  multiplication 
law  to  a^. 

Hence  by  definition  a^  is  the  square  root  of  a^. 
The  denominator  of  a  fractional  exponent  indicates  that 
a  root  is  to  be  found,  while  the  numerator  denotes  a  power. 

EXERCISES 

1.  Find  the  value  of:  4^  25\  4%  16%  36^  4%  2^  2=,  8^ 
8t,  30X2-^X49^  2X50X10-2X36^  8*,   8*,  8%  4"^. 

2.  Express  with  fractional  exponents:     Vo  ,  V3  ,  Vo^ ,  V2a'. 

3.  Express  without    fractional    exponents:    a*,   a=,    a^b^y   2a'b^ 

228.  Usefulness.  Fractional  and  negative  exponents  arc 
used  very  frequently  in  higher  mathematics.  In  fact,  their 
invention  made  great  advances  in  mathematics  possible. 
If  you  go  on  to  the  further  study,  of  mathematics,  you  will 
find  that  there  are  still  other  kinds  of  numbers  that  can 
be  used  as  exponents. 

APPENDIX  D 


lABLE  OF  SQUARE  ROOTS  OF  NUMBERS  FROM  0  TO  '99 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0.000 

1.000 

1.414 

1.732 

2 .  000 

2.236 

2.449 

2.646 

2.828 

3.000 

1 

3.162 

3. .317 

3.464 

3.606 

3.742 

3.873 

4.000 

4.123 

4.243 

4.359 

2 

4.472 

4.583 

4.690 

4.796 

4.899 

5.000 

5.099 

5.196 

5.292 

5.385 

3 

5.477 

5 . 5()8 

5.657 

5.745 

5.831 

5.916 

6.000 

6.083 

6.164 

6.245 

4 

6.325 

6.403 

6.481 

6 .  557 

6 .  ()33 

6.708 

6.782 

6.856 

6.928 

7.000 

5 

7.071 

7.141 

7.211 

7.280 

7.348 

7.416 

7.483 

7.550 

7.616 

7.681 

6 

7.746 

7.810 

7.874 

7.937 

8.000 

8.062 

8.124 

8.185 

8.240 

8.307 

7 

8.367 

8.426 

8.485 

8.544 

8.602 

8.660 

8.718 

8.775 

8.832 

8.88S 

8 

8.944 

9.000 

9.055 

9.110 

9.165 

9.220 

9.274 

9.327 

9.381 

9.434 

9 

9.487 

9.539 

9.592 

9.644 

9.695 

9.747 

9.798 

9.849 

9.894 

9.950 

THE  INDEX 

[Bold-face  figures  indicate  an  extended  treatment  of  the  subject  indexed. 
The  numbers  refer  to  pages.] 


Absolute  term,  155 

Absolute  value  of  a  number,  73 

Accuracy: 

algebraic  and  graphic  solutions 
compared,  126 

approximate  numbers,  13 

in  calculations,  16 
Addition: 

algebraic,  72,  73 

law  of  signs,  73 

of  decimals,  9 

of  fractions,  5,  214,  218 

of  polynomials,  183 

of  positive  and  negative  num- 
bers, 72 

of  radicals,  274 

of  terms,  183 

on  number  scale,  71 
Ahmes,  67 
Angle: 

complement,  206 

supplement,  206 
Angles,  sum  of,  in  a  triangle,  49 
Antecedent,  definition,  225 
Approximate  numbers: 

in  solution  of  quadratic  equa- 
tion, 237,  284 

use  of,  13 

value  of  square  roots,  23 
Arabs,  56,  209 
Area,  formulas  for: 

circle,  241 

cylinder,  177 

rectangle,  20,  23 

trapezoid,  179 

triangle,  159 
Arithmetic,  fractions  and  decimals 

of,  1  ff. 
Aryabhatta,  68 
Ascending  powers,  194 
i\xis: 

in  graphs,  119 

:x;-axis,  >'-axis,  120 


Babylonians,  56,  208 


Balance: 

of  equations,  29 

pan,  28 
Bar,  use  of,  189 
Binomial : 

definition,  139,  181 

product  of  binomial  and  mono- 
mial, 143 

product  of  two  binomials,  151  ff., 
161 

square  of,  165 
Braces,  use  of,  189 
Brackets,  use  of,  189 

Calipers,  15 
Checking : 

of  equations,  30 

of  identities,  148 
Circle: 

area,  formula  for,  241 

circumference,  13  ff. 
Clearing  of  fractions,  245 
Coefficient,  definition,  30 
Completing  the  square,  234 
Conjugate,  definition,  281 
Consequent,   definition,   225 
Constants,  definition,  227 
Coordinates,  definition,  118 
Cross-ruled  paper,  use  of,  102 
Cylinder,  formula  for  area,  177 

Decimals,  8  fif. 

addition  and  subtraction  of,  9 

multiplication  and  division  of, 
9ff. 
Degree : 

of  integral  expressions,  182 

of  terms,  147,  182 
Denominator,  least  common,  216 
Descartes,  120 
Descending  powers,  194 
Diagram,    bar    and    fine.     See 

Graphs 
Digit  problems,  51 
Divisibility  tests,  142 


299 


300 


THE  INDEX 


Division : 
as  a  fraction,  196 
by  zero,  40 
inexact,  195 

law  of  exponents  in,  141,  191 
law  of  signs  in,  92 
of  decimals,  9  f.,  12 
of  fractions,  4,  213 
of  polynomials,  193 
of  positive  and  negative  num- 
bers, 92 
of  radicals,  279 
of  terms,  94,  192 

Egyptians,  55,  67,  209 
Elimination : 
by  addition,  126 
by  substitution,  130 
Ellipse,  266 
Equations,  20  ff. 
checking  of,  30 
clearing  of,  245 
definition,  26 
dependent,  258 

factorable,  in  one  unknown,  197 
fractional,  245  ff. 
general,  58 
inconsistent,  258 
independent,  258 
in  two  unknowns,  267  ff. 
irrational,  282 

linear.     See  Linear  equations 
literal,  59 
numerical,  59 
operations  on.     See  Operations 

on  equations 
quadratic.   See  Quadratic  equa- 
tions 
root  of,  30,  150 
sets  of,   algebraic  solution  of, 
126,  262  ff.    See  also  Sets  of 
equations 
solution  of : 

by  factoring,  149  ff. 
by  substitution,  130 
graphic.  111 
Equivalent  forms,  144 
Expanded  form,  definition,  172 
Exponents: 
fractional,  298 
integral,  138 
law  of: 

division,  141 


multiplication,  140 
negative,  297 

positive  integer,  191,  296 
zero,  191,  297 

Factor: 

common,  145 

definition,  138 

prime,  142 

zero,  149 
Factored  form,  145 
Factoring,  142  ff. 

after  expanding,  290 

by  grouping,  288 

common  binomial  factor,  288 

common  factor,  145 

difference  between  two  cubes, 
287 

difference  between  two  squares, 
170,  289 

identities,  use  of,  172 

numbers,  142 

possibility  of,  174 

sum  of  two  cubes,  287 

trinomial,  155,  161 

types,  summary  of,  172 
First  degree,  expressions  of,  147 
Formulas,  20  ff. 

evaluation  of,  63 

transformation  of,  248 
Fractional  equation,  245 
Fractions,  208  ff. 

addition  of,  5,  214,  218 

clearing  equations  of,  245 

complex,  221 

definitions,  208 

division  of,  4,  213 

equal,  1 

fundamental  principle  of,  1,  209 

graphs  of,  224 

history  of,  208 

laws  governing  use  of,  209 

multiplication  of,  2,  3,  211 

of  arithmetic,  1  ff. 

reduction  to  lowest  terms,  1, 210 

signs,  219 

square  root  of,  232 

subtraction  of,  5,  214 
Fundamental  operations,    review 
and  extension  of,  181  ff. 

General  equation,  58 
General  number,  55 


THE   INDEX 


301 


Generalized  problems,  60 
Generalized  statements,  55 
Graphics,  101  ff.     See  also  Graphs 
Graphic  solution : 

of  linear  and  quadratic  sets,  264 

of  linear  sets,  120,  124,  263 

of  problems,  115 
Graphs: 

algebraic  expression,  117 

bar,  101 

comparison  of,  108 

definition,  105 

line,  105 

linear,  125 

mechanical,  109 

of  fractions,  224 

of  quadratics,  265 

problems  solved  by,  115 
Greeks,  55,  208 

Hindus,  56,  65,  68,  209 
Hyperbola,  266 
Hypotenuse,  formula  for,  242 

Identities: 

checking  work  by  substitution 
in,  148 

summary  of,  172 

use  of,  172 
Identity,  definition,  148 
Inconsistent  equations,  257 
Independent  equations,  258 
Integral    expressions,    degree   of, 

182 
Irrational  equations,  282 
Irrational  numbers,  273 
Irrational  foots,  231 

Letters,  use  of,  26,  56 
Lever,  206 

Like  or  similar  terms,  182 
Line  graphs,  105 
Linear  equations : 

algebraic  solution  of  set: 
by  addition,  126  ff. 
by  substitution,  130 

definition,  125 

graphic  solution  of  a  set,  120, 
124,  263 

in  two  unknowns,  123  fif. 

standard  form,  129 
Linear  expressions,  definition,  147 
Lowest  common  denominator,  216 
Lowest  common  multiple,  216, 217 


Lowest  terms,  reduction,  of  frac- 
tion to,  1,  210 

Measuring,  numbers  obtained  by, 

15 
Mixed  numbers,  operations  with,  6 
Monomial: 

definition,  139,  181 

product  of  binomial  and,  143 

products  of  monomials,  139  f. 

quotient  of  monomials,  140  f. 
Multiple: 

common,  215 

definition,  215 

least  common,  216,  217 
Multiplication : 

law  of  exponents,  191 

law  of  signs,  89 

of  decimals,  9  ff, 

of  fractions,  2,  23,  211 

of  polynomials,  90,  185 

of  positive  and  negative  num- 
bers, 89 

of  radicals,  278 

Negative,  double  use  of  word,  84 
Negative  numbers,  65  ff. 

addition  of,  73 

definition,  66 

division  of,  92 

graphing  of,  119 

multiplication  of,  89 

subtraction  of,  79 

sum  of  positive  and,  73,  76 

usefulness  of,  66 
Numbers: 

approximate,  13 

irrational,  273 

kinds  of,  67,  273 

mixed,  operations  with,  6 

negative.  See  Negative  numbers 

obtained  by  measuring,  15 

positive,  66,  69 

prime,  142 
Number  scale,  68 
Number  symbols,  55 
Number  system,  67 

Operations  on  equations,  27 
addition,  36 
division,  29 

fundamental,  review  and  exten- 
sion of,  181  ff. 


302 


TrfE   INDEX 


Operations  on  equations  (continued) 

multiplication,  37 

square  root,  233 

subtraction,  32 

summary  of,  40 
Order  of  operations,  18 


7r,13 

Parabola,  266 
Parentheses: 

fractions  as,  96 

insertion  of  expressions  in,  190 

removal  of,  189 

use  of,  19,  27,  189 

within  parentheses,  189 
Polynomials: 

addition  of,  183 

definition,  181 

division  of,  193 

multiplication  of,  90,  185 

square  root  of,  291,  292 

subtraction  of,  188 
Positive  numbers,  66,  69 
Powers : 

ascending  and  descending,  194 

cube,  138 

definition,  138 

square,  138 
Problems : 

algebraic  solution  of,  23,  42  ff. 

graphic  solution  of,  115 

impossible,  98 

solution  by  one  unknown,  44 

solution  by  two  unknowns,  132 

statement  of,  47,  199  ff. 

with  impossible  answers,  98 
Problems  to  solve: 

circle,  177,  241 

coin,  53,  99 
'   cylinder,  177,  242 

digit.     See  Number,  below 

interest,  205 

mixture,  100,  135,  267 

number,  44,  62,  87,  99, 134, 179, 
201,  251,  269 
consecutive  integers,  52,  88, 

99,  160,  164,  201,  269 
digit,  51,  134,  164,  202,  269 

per  cent,  24 

ratio,  8,  225  f . 

rectangle,  48,  87,  88,  159,  160, 
164.  179.  204.  243 


speed,  25,  54,  113, 115, 116,  202, 

256,  268 
tank,  55,  113,  253,  271 
teeter,  206,  268 
trapezoid,  179,  180 
triangle,    49,  62,  87,  88,   159, 

160,  206 

right,  242,  243 

similar,  251 
uniform    motion.     See    Speed, 

above 
work,  253,  254,  255,  271 
Product: 

of  binomial  and  monomial,  143 
of  monomials,  139  f. 
of  sum  and  difference,  168 
of  two  binomials,  151  ff.,  161 
Product  form,  change  of,  to  sum 

form,  144 
Products,  special,  137  ff.,  165  f. 

Quadratic  equations,  232  fif. 
definition,  147,  197,  232 
graph  of,  239 
roots  of,  150 
solution  of: 

by  completing  the  square,  234 

by  factoring,  149 

by  square  root,  233 

graphic,  239 

of  sets,  262 

possibility  of  solving  all,  238 
square  roots  and,  229  ff. 
standard  form,  234 

Radicals,  272  ff. 

addition  of,  274 

definition,  273 

division  of,  279 

evaluation  of,  274 

first  use  of  sign,  229 

like,  275 

multiplication  of,  278 

simple,  275 
Ratio,  7,  225 
Rationalizing,  280 
Reciprocal,  4 
Rectangle,  formula  for  area  of, 

20,23 
Riese,  229 

Right  triangle,  formula  for  hypot- 
enuse, 242 
Roentgen,  27 


THE  INDEX 


303 


Root  of  equation,  30,  150 

irrational,  231 
Roots.    See  Square  roots 
Rudolff ,  229 

Second    degree,    expressions    of, 

147,  182 
Sets  of  equations: 

algebraic  solution  of,  126  ff.,  262 

fractional,  260 

graphic  solution  of,  124,  264 

Unear,  124,  258 

linear-quadratic,  264 
Signs: 

double  meaning  of,  75 

laws  for: 
addition,  73 
division,  92 
multiplication,  88 
subtraction,  79 

of  fractions,  219 

+,— ,  origin  of,  66 

radical,  30,  229 
Similar  or  like  terms,  182 
Solution  of  equations.     See  Equa- 
tions; Linear  equations;  Quad- 
ratic equations 
Square  of  binomial,  165 
Square  of  terms,  140 
Square  root,  229  flf. 

by  division,  294 

by  trial,  230 

definition,  167 

of  fractions,  232 

of  numbers,  231,  272,  294 

of  polynomials,  291,  292 

of  terms,  272 
Square  roots: 

and  quadratic  equations,  229  ff. 

approximate  values  of,  231 

number  of,  231 

table  of,  298 
Squares,  table  of,  230 
Standard  form: 

of  linear  equation  in  two  un- 
knowns, 129 

of  quadratic  equation,  234 
Stating  problems,  47,  200 
Stevin,  68 
Straight  Une,  125 
Subtraction: 

algebraic,  78 

of  decimals,  9 


of  polynomials,  188 
Stmi  form,     change    to    product 
form,  144 

Teeter  problems,  206,  268 
Temperature  graph,  106,  110 
Term: 

absolute,  155 

definition,  30,  181 
Terms : 

addition  of,  77,  183 

definition,  30,  181 

degree  of,  147,  182 

kinds  of,  181 

like  or  similar,  182 

rationalizing  of  terms  of  a  frac- 
tion, 280 

square  of,  140 

square  root  of,  272 
Thermograph,  110 
Thermometer,  68 
Translations,  42 

from  algebra  into  English,  exer- 
cises in,  43,  57 

from  English  into  algebra,  exer- 
cises in,  42,  57 
Trapezoid,   formula  for  area  of, 

180 
Triangle,  49 

area  of,  formula  for,  159 

right,  242 

similar,  251 
Trinomial : 

definition,  139 

factoring,  155  ff. 

factors  of  quadratic,  162 

general  quadratic,  161 

Uniform  motion,  200 
Unknown,  definition,  26,  30 

Variable,  117,  227 
Variation,  227 
Vieta,  82 

Widmann,  66 

X-axis,  120 

y-axis,  120 

Zero: 

division  by,  40 
exponent,  191,  297 
factor,  149 


YB  35917 


571787 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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